\documentclass[11pt]{article} \usepackage{enumerate, amsmath, amsthm, amsfonts, amssymb, xy, color, mathrsfs, graphicx, paralist} \usepackage[margin=1in]{geometry} \usepackage{hyperref} \usepackage[inline]{showlabels} \usepackage{ytableau} \input xy \xyoption{all} \numberwithin{equation}{section} \newtheorem{theorem}[equation]{Theorem} \newtheorem*{thm}{Theorem} \newtheorem{proposition}[equation]{Proposition} \newtheorem{lemma}[equation]{Lemma} \newtheorem{corollary}[equation]{Corollary} \newtheorem{conjecture}[equation]{Conjecture} \newtheorem{problem}[equation]{Problem} \theoremstyle{definition} \newtheorem{rmk}[equation]{Remark} \newenvironment{remark}[1][]{\begin{rmk}[#1] \pushQED{\qed}}{\popQED \end{rmk}} \newtheorem{eg}[equation]{Example} \newenvironment{example}[1][]{\begin{eg}[#1] \pushQED{\qed}}{\popQED \end{eg}} \newtheorem{defn}[equation]{Definition} \newenvironment{definition}[1][]{\begin{defn}[#1]\pushQED{\qed}}{\popQED \end{defn}} % Common number systems \newcommand{\bC}{\mathbf{C}} \newcommand{\N}{\mathbf{N}} \newcommand{\Q}{\mathbf{Q}} \newcommand{\bR}{\mathbf{R}} % Preferences \renewcommand{\phi}{\varphi} \renewcommand{\emptyset}{\varnothing} \newcommand{\eps}{\varepsilon} \newcommand{\injects}{\hookrightarrow} \newcommand{\surjects}{\twoheadrightarrow} \renewcommand{\tilde}[1]{\widetilde{#1}} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\DS}{\displaystyle} \newcommand{\leftexp}[2]{\vphantom{#2}^{#1} #2} \newcommand{\steven}[1]{\textcolor{red}{$[\star$ Steven: #1 $\star]$}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{{\tt arXiv:#1}}} \makeatletter \def\Ddots{\mathinner{\mkern1mu\raise\p@ \vbox{\kern7\p@\hbox{.}}\mkern2mu \raise4\p@\hbox{.}\mkern2mu\raise7\p@\hbox{.}\mkern1mu}} \makeatother % Algebra \DeclareMathOperator{\im}{image} %image \DeclareMathOperator{\codim}{codim} %codimension \DeclareMathOperator{\coker}{coker} %cokernel \renewcommand{\hom}{\operatorname{Hom}} %hom-set \newcommand{\cHom}{\mathcal{H}om} \DeclareMathOperator{\supp}{Supp} %support \DeclareMathOperator{\rad}{rad} %radical \DeclareMathOperator{\ass}{Ass} %associated primes \DeclareMathOperator{\ann}{Ann} %annihilator \DeclareMathOperator{\trace}{Trace} %trace \DeclareMathOperator{\norm}{Norm} %norm \DeclareMathOperator{\nil}{nil} %nilradical \DeclareMathOperator{\rank}{rank} %rank \DeclareMathOperator{\ext}{Ext} %Ext \DeclareMathOperator{\reg}{reg} %Castelnuovo-Mumford regularity \newcommand{\GL}{\mathbf{GL}} \newcommand{\SL}{\mathbf{SL}} \newcommand{\Sp}{\mathbf{Sp}} \newcommand{\SO}{\mathbf{SO}} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Pf}{Pf} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Der}{Der} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\depth}{depth} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\pdim}{pdim} \DeclareMathOperator{\idim}{idim} \DeclareMathOperator{\gldim}{gldim} \DeclareMathOperator{\irank}{irank} \DeclareMathOperator{\ad}{ad} \DeclareMathOperator{\Spec}{Spec} \newcommand{\bA}{\mathbf{A}} \newcommand{\cA}{\mathcal{A}} \newcommand{\sA}{\mathscr{A}} \newcommand{\bB}{\mathbf{B}} \newcommand{\cB}{\mathcal{B}} \newcommand{\bE}{\mathbf{E}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\rh}{\mathrm{h}} \newcommand{\rH}{\mathrm{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\bK}{\mathbf{K}} \newcommand{\cK}{\mathcal{K}} \newcommand{\fm}{\mathfrak{m}} \newcommand{\cM}{\mathcal{M}} \newcommand{\fn}{\mathfrak{n}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\bP}{\mathbf{P}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\bS}{\mathbf{S}} \newcommand{\fS}{\mathfrak{S}} \newcommand{\bT}{\mathbf{T}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cX}{\mathcal{X}} \newcommand{\fX}{\mathfrak{X}} \newcommand{\bY}{\mathbf{Y}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\bZ}{\mathbf{Z}} \newcommand{\cZ}{\mathcal{Z}} \DeclareMathOperator{\grade}{grade} \DeclareMathOperator{\mult}{mult} \DeclareMathOperator{\pf}{pf} \DeclareMathOperator{\ind}{ind} \DeclareMathOperator{\res}{res} \newcommand{\Gr}{\mathbf{Gr}} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\diag}{diag} \newcommand{\MU}{\boldsymbol\mu} \newcommand{\F}{\mathbf{F}} \DeclareMathOperator{\Jac}{Jac} \DeclareMathOperator{\Pic}{Pic} \newcommand{\cL}{\mathcal{L}} \renewcommand{\i}{\mathbf{i}} \newcommand{\spin}{\mathbf{spin}} \newcommand{\Spin}{\mathbf{Spin}} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Conf}{Conf} \DeclareMathOperator{\Prob}{Prob} \title{Gibbs measures on branching graphs, lectures by A. Borodin} \author{typed by Steven Sam} \begin{document} \maketitle %\section{Lecture 1, 9/8/2011} \section{Introduction} \begin{example}[Pascal's triangle] Consider the 2-d grid $\N^2$ graded by diagonal, i.e., the $n$th level is $\Gamma_n = \{(k,\ell) \mid k+\ell =n\}$. A path is a sequence of vertices that increases one in level at each step such that adjacent vertices are connected by an edge of the form $(0,1)$ or $(1,0)$. %\begin{center} % \includegraphics[height=5cm]{lec1pic1} %\end{center} We will consider measures on the set of all paths. A measure is a Gibbs measure if for any fixed $(k,\ell)$, the probability of a given initial segment going from $(0,0)$ to $(k,\ell)$ appearing in a random path is the same probability as any other initial segment. Problems: 1. What are all Gibbs measures? There is a notion of convex combination of Gibbs measures, so we want to describe the extremal (ergodic) ones. In many of our cases, our set of Gibbs measures will be a simplex. 2. How to decompose a ``natural'' Gibbs measure into extremal ones? For a point $(k,\ell)$, let $m(k,\ell)$ denote the probability of a path passing through $(k,\ell)$. Then $\sum_{k + \ell = n} m(k,\ell) = 1$ by definition. Furthermore, \[ m(k,\ell) = \frac{\binom{k+\ell}{\ell}}{\binom{k+\ell+1}{k+1}} m(k+1,\ell) + \frac{\binom{k+\ell}{\ell}}{\binom{k+\ell+1}{\ell+1}} m(k,\ell+1). \] To prove this, we just need to note that the number of paths from $(0,0)$ to $(k,\ell)$ using just the steps $(1,0)$ and $(0,1)$ is $\binom{k+\ell}{\ell}$ because each path will have exactly $k+\ell$ steps, and we are just choosing some subset of size $\ell$ to be steps of the form $(0,1)$. Denote $\phi(k,\ell) = m(k,\ell) / \binom{k+\ell}{\ell}$. Then we want to find all solutions to the equation \[ \phi(k,\ell) = \phi(k+1,\ell) + \phi(k,\ell+1) \] where $\phi \ge 0$ and $\phi(0,0) = 1$. One possible solution is $\phi(k,\ell) = 2^{-k-\ell}$. A more general one is to take $\phi(k,\ell) = p^k(1-p)^\ell$ with $0 \le p \le 1$. These give all the extremal ones. \end{example} \[ \ytableausetup{smalltableaux} {\small \xymatrix{ & & \ydiagram{3} \ar[r] \ar[dr] & \ydiagram{4}\\ & \ydiagram{2} \ar[ur] \ar[dr] & & \ydiagram{3,1}\\ \ydiagram{1} \ar[ur] \ar[dr]& & \ydiagram{2,1} \ar[ur] \ar[r] \ar[dr] & \ydiagram{2,2}\\ & \ydiagram{1,1} \ar[ur] \ar[dr] & & \ydiagram{2,1,1}\\ & & \ydiagram{1,1,1} \ar[r] \ar[ur] & \ydiagram{1,1,1,1} } } \] \begin{example}[Young's graph] Classification of Gibbs measures is equivalent to the following question: Find all sequences $(\lambda(n))_n$ of Young diagrams (where $|\lambda(n)|=n$) such that \[ \lim_{n \to \infty} \frac{\chi^{\lambda(n)}}{\dim \lambda(n)} \] exists (interpreted via the tower of inclusions $S_1 \subset S_2 \subset S_3 \subset \cdots$ of symmetric groups). Equivalently, find all sequences $(c_k)_{k \ge 1}$ such that the infinite Toeplitz matrix \[ \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots \\ 0 & 1 & c_1 & c_2 \\ 0 & 0 & 1 & c_1 \\ 0 & 0 & 0 & 1\\ \vdots \end{pmatrix} \] is totally positive, i.e., all finite minors are positive. \end{example} \begin{example} We can also consider the tower of inclusions $U(1) \subset U(2) \subset U(3) \subset \cdots$ of unitary groups. Now we consider its branching graph instead of Young's graph. The question about Gibbs measures ends up being related to characterizing sequences $(c_k)_{k \in \bZ}$ such that the infinite Toeplitz matrix \[ \begin{pmatrix} c_0 & c_1 & c_2 & c_3 & \cdots \\ c_{-1} & c_0 & c_1 & c_2 \\ c_{-2} & c_{-1} & c_0 & c_1 \\ c_{-3} & c_{-2} & c_{-1} & c_0\\ \vdots \end{pmatrix} \] is totally positive. \end{example} \begin{example} We can also consider the graph of eigenvalues of Hermitian matrices. The $N$th level is \[ \Gamma_N = \{\lambda_1 \ge \cdots \ge \lambda_N \} \subset \bR^N \] The restriction is defined by taking the upper left $(N-1) \times (N-1)$ submatrix of an $N \times N$ Hermitian matrix $H$. If $H$ is Hermitian with eigenvalues $\lambda_1 \ge \cdots \ge \lambda_N$ and $H'$ is its restriction with eigenvalues $\mu_1 \ge \cdots \ge \mu_{N-1}$, then we have the interlacing condition $\lambda_1 \ge \mu_1 \ge \lambda_2 \ge \mu_2 \ge \lambda_3 \ge \cdots \ge \mu_{N-1} \ge \lambda_N$. This is related to characterizing functions $f \colon \bR \to \bR$ such that we have $\det(f(x_i-y_j))_{i,j=1}^N \ge 0$ for all $x_1 < \cdots < x_N$ and $y_1 < \cdots < y_N$. \end{example} \begin{example} Let $\bZ$ be the 1-dimensional lattice. There are some particles on this lattice. After each unit of time, this system evolves as follows: for a given particle, if its space to the right is full, nothing happens. If it is empty, then it moves to the right with probability $1/2$. One can imagine this being a simple model for cars driving on the highway. Basic question: what happens after a large amount of time? If we start with a single particle and observe its behavior. Then this becomes the central limit theorem. The distance it travels in time $t$ is $N(0,1) t^{1/2}$. If we start with one particle at each even integer, then the distance travelled of a single particle is $vt + ct^{1/3}$ for some coefficients $v,c$, where $c$ is NOT a normal distribution and $v$ is easy to calculate. \end{example} \section{Background} \subsection{Character theory of the symmetric group} A {\bf partition} of $n$ is $\lambda_1 + \lambda_2 + \cdots = n$ with $\lambda_1 \ge \lambda_2 \ge \cdots \ge 0$. We represent them graphically using Young diagrams. Denote the set of partitions of $n$ by $\bY_n$. For a partition $\lambda$, we use $\lambda'$ to denote the conjugate partition. We can also represent $\lambda$ using Frobenius notation $(\lambda_1 - 1, \lambda_2 - 2, \dots | \lambda'_1 - 1, \lambda'_2 - 2, \dots) = (p_1, \dots, p_d | q_1, \dots, q_d)$. Then $\sum_{i=1}^d (p_i+q_i+1) = n$. For $(i,j) \in \lambda$, we use $h(i,j)$ to denote the hook length. We use $S_n$ to denote the symmetric group on $n$ symbols. \begin{compactenum}[\rm 1.] \item The conjugacy classes of $S_n$ are parametrized by $\bY_n$ via cycle lengths. \item The set of irreducible complex representations of $S_n$ is also in ``natural'' bijection with $\bY_n$. For $\lambda \in \bY_n$, we use $\pi_\lambda \colon S_n \to \GL(V_\lambda)$ to denote the corresponding representation. We will write $\dim \lambda = \dim V_\lambda$. \item Hook length formula: \begin{align*} \frac{\dim \lambda}{n!} &= \frac{1}{\prod_{(i,j) \in \lambda} h(i,j)} = \det\left(\frac{1}{(\lambda_i - i + j)!}\right)_{i,j=1}^n\\ &= \frac{\prod_{i N \end{cases}. \] We define \[ \Lambda^k = \varprojlim_N \Lambda^k_N = \{(f_1, f_2, f_3, \dots) \mid f_j \in \Lambda_j^k, \ \pi_j f_j = f_{j-1} \text{ for all } j\}, \] which we call the space of homogeneous symmetric functions of degree $k$. We also set $\Lambda = \bigoplus_{k \ge 0} \Lambda^k$ the algebra of symmetric functions. \subsubsection{Monomial symmetric functions} The {\bf monomial symmetric polynomials} are $m_\alpha(x) = \sum_\beta x^\beta$ where the sum is over all unique rearrangements $\beta$ of $\alpha = (\alpha_1, \dots, \alpha_N)$. For notation, we will usually require that $\alpha$ is a partition, and we will write $\lambda$ usually. It is clear that the $m_\lambda$ form a basis of $\Lambda_N$ as we index over all partitions with at most $N$ parts. For any partition $\lambda$, the sequence $(f_j = m_\lambda(x_1, \dots, x_j))$ (with the understanding that $f_j = 0$ if $\ell(\lambda) > j$) is a symmetric function, which we will denote by $m_\lambda = m_\lambda(x_1, x_2, \dots)$. \begin{example} The function $f_j(x_1, x_2, \dots) = \prod_{i = 1}^j (1 + x_i)$ satisfies $\pi_jf_j = f_{j-1}$, but it is not a symmetric function because it is not a finite linear combination of its homogeneous components. \end{example} \subsubsection{Elementary symmetric functions} Define \[ e_k = m_{(1,\dots,1)} = \sum_{i_1 < i_2 < \cdots < i_k} x_{i_1} \cdots x_{i_k} \] and $e_0 = 1$. Then we have \[ E(t):=\sum_{k \ge 0} e_k t^k = \prod_{i \ge 1} (1+x_it). \] For any partition $\lambda$, we define the {\bf elementary symmetric functions} $e_\lambda = e_{\lambda_1} e_{\lambda_2} \cdots$. The {\bf lexicographic order} $<$ on $\bY_n$ is defined by comparing partitions term-by-term and comparing the first non-equal entries. \begin{proposition} Let $\lambda \in \bY_n$ and let $\lambda'$ be its conjugate. Then $e_{\lambda'} = m_\lambda + \sum_{\mu < \lambda} a_{\lambda,\mu} m_\mu$ where $a_{\mu,\lambda} \in \N$. \end{proposition} \begin{proof} The coefficients of the decomposition of any symmetric function on $\{m_\lambda\}$ are just the coefficients in front of suitable monomials (namely, we look at those whose exponents form a partition). We have \[ e_{\lambda'} = e_1^{\lambda_1 - \lambda_2} e_2^{\lambda_2 - \lambda_3} \cdots = (\sum_i x_i)^{\lambda_1 - \lambda_2} (\sum_{i < j} x_ix_j)^{\lambda_2 - \lambda_3} \cdots \] The highest monomial in lexicographic order is $x_1^{\lambda_1 - \lambda_2 + \lambda_2 - \lambda_3 + \cdots} x_2^{\lambda_2 - \lambda_3 + \lambda_3 - \lambda_4 \cdots} \cdots$. So we see that $e_{\lambda'} = m_\lambda + \text{lower order terms}$. \end{proof} As a corollary, we get the following theorem. \begin{theorem}[Fundamental theorem of the theory of symmetric functions] $\Lambda = \bR[e_1, e_2, \dots]$, i.e., the $e_\lambda$ form a basis for $\Lambda$. \end{theorem} \subsubsection{Complete homogeneous symmetric functions} Define $h_0 = 1$ and for $k > 0$, define \[ h_k = \sum_{|\lambda| = k} m_\lambda = \sum_{i_1 \le \cdots \le i_k} x_{i_1} \cdots x_{i_k}. \] Then \[ H(t):=\sum_{k \ge 0} h_k t^k = \prod_{i \ge 1} (1-x_it)^{-1} = \prod_{i \ge 1} (1 + x_it + (x_it)^2 + \cdots). \] Also for any partition $\lambda$ define $h_\lambda = h_{\lambda_1} h_{\lambda_2} \cdots$. \begin{proposition} $\Lambda = \bR[h_1, h_2, \dots]$, i.e., the $h_\lambda$ form a basis for $\Lambda$. \end{proposition} \begin{proof} Note that $E(t) H(-t) = 1$. This is equivalent to the conditions \[ \sum_{r=0}^n (-1)^r e_r h_{n-r} = 0 \] for all $n>0$. Since we already know that $\Lambda = \bR[e_1, e_2, \dots]$, we can define an algebra homomorphism $\omega \colon \Lambda \to \Lambda$ by $\omega(e_k) = h_k$. From the relations above, we see that $\omega(h_k) = e_k$, so the $h_\lambda$ are linearly independent, as desired. \end{proof} \subsubsection{Newton power sum symmetric functions} Define $p_0 = 1$ and \[ p_n = \sum_{i \ge 1} x_i^n = m_{(n)} \] for $n > 0$. Then we have \begin{align*} P(t) &= \sum_{r \ge 1} \frac{p_r t^r}{r} = \sum_{i \ge 1} \sum_{r \ge 1} \frac{x_i^r t^r}{r} = \sum_{i \ge 1} - \ln(1-x_it) = -\ln \prod_{i \ge 1} (1-x_it) = \ln H(t). \end{align*} For a partition $\lambda$, define $p_\lambda = p_{\lambda_1} p_{\lambda_2} \cdots$. \begin{proposition} $\Lambda = \bR[p_1, p_2, \dots]$, i.e., the $p_\lambda$ is a basis for $\Lambda$. \end{proposition} \begin{remark} The corresponding result for the $e_\lambda$ and $h_\lambda$ holds with $\bZ$-coefficients, but for $p_\lambda$, one has to use $\Q$-coefficients. \end{remark} \begin{proof} We have \[ \sum_{r \ge 1} p_r t^{r-1} = P'(t) = H'(t) / H(t), \] which translates to $nh_n = \sum_{r=1}^n p_r h_{n-r}$. Similarly, $P'(-t) = E'(t) / E(t)$, which translates to $ne_n = \sum_{r=1}^n (-1)^{r-1} p_r e_{n-r}$. \end{proof} \subsubsection{Schur functions} For a monomial $x_1^{\alpha_1} \cdots x_N^{\alpha_N}$, define its antisymmetrization as $\sum_{\sigma \in S_N} \sgn(\sigma) x_{\sigma(1)}^{\alpha_1} \cdots x_{\sigma(N)}^{\alpha_N}$. Note multiplication by $\prod_{i < j} (x_i - x_j)$ is a bijection between symmetric and skew-symmetric polynomials in $x_1, \dots,x_N$ (boson--fermion correspondence). Let $\lambda = (\lambda_1, \dots, \lambda_N)$ be a partition. Define $\delta = (N-1,N-2,\dots,1,0)$. Then $\lambda+\delta=(\lambda_1+N-1, \dots ,\lambda_{N-1}+1, \lambda_N)$ is a strict partition. We will set $a_{\lambda+\delta}$ to be the antisymmetrization of $x^{\lambda+\delta}$. Then we define the {\bf Schur polynomial} $s_\lambda = a_{\lambda+\delta} / a_{\delta}$. \begin{proposition} $\{s_\lambda\}_{\ell(\lambda) \le N}$ form a basis in $\Lambda_N = \bR[x_1, \dots, x_N]^{S_N}$. \end{proposition} \begin{proof} The $a_{\lambda + \delta}$ form a basis for the skew-symmetric polynomials. The rest follows from the boson--fermion correspondence. \end{proof} By convention, we set $s_\lambda(x_1, \dots, x_N) = 0$ if $\ell(\lambda) > N$. \begin{lemma} $s_\lambda(x_1,\dots,x_N,0) = s_\lambda(x_1, \dots, x_N)$. Hence $s_\lambda \in \Lambda$, which we call the {\bf Schur symmetric function}. \end{lemma} \begin{proposition}[Cauchy's identity] \[ \sum_{\lambda \in \bY} s_\lambda(x) s_\lambda(y) = \prod_{i,j} \frac{1}{1-x_iy_j}. \] \end{proposition} \begin{proof} We will use Cauchy's determinant, which says that \[ \det \left( \frac{1}{1-x_iy_j} \right)_{i,j=1}^N = \frac{\prod_{i k_2 > \cdots > k_N \ge 0} \sum_{\sigma, \tau S_N} \sgn(\sigma) (x_{\sigma(1)} y_{\tau(1)})^{k_1} \cdots (x_{\sigma(N)} y_{\tau(N)})^{k_N}. \qedhere \] \end{proof} \begin{proposition}[Jacobi--Trudi] For any $\lambda \in \bY$, \[ s_\lambda = \det(h_{\lambda_i - i + j})_{i,j=1}^N \] with the convention that $h_m=0$ for $m<0$. \end{proposition} \begin{proof} General fact: if $f(u) = \sum_{m \ge 0} f_m u^m$, then \[ f(x_1) \cdots f(x_N) = \sum_{\lambda:\ell(\lambda) \le N} \det(f_{\lambda_i-i+j})_{i,j=1}^N s_\lambda(x_1, \dots, x_N). \] To see this, multiply both sides by $a_\delta(x)$ to get \[ a_\delta(x_1, \dots, x_N) f(x_1) \cdots f(x_N) = \sum_{\lambda:\ell(\lambda) \le N} \det(f_{\lambda_i-i+j})_{i,j=1}^N a_{\lambda+\delta}(x_1, \dots, x_N). \] To obtain the coefficient of $a_{\lambda + \delta}(x_1, \dots, x_N)$ in the left-hand side, we need extract the coefficient of $x_1^{\lambda_1+N-1} \cdots x_N^{\lambda_N}$. But the left-hand side is just \[ \sum_{\tau \in S_N} \sgn(\tau) x_1^{N-\tau(1)} \cdots x_N^{N-\tau(N)} (\sum f_{m_1}x_1^{m_1}) \cdots (\sum f_{m_N} x_N^{m_N}), \] which implies the fact. To use this fact, consider \[ f(u) = \sum h_m(y_1, \dots, y_N) u^m = \prod_j \frac{1}{1-y_ju}. \] By the Cauchy identity, we have $f(x_1) \cdots f(x_N) = \prod_{i,j} (1-x_iy_j)^{-1}$, so we are done. \end{proof} \begin{proposition}[Giambelli] $s_\lambda = \det(s_{(p_i|q_j)})_{i,j=1}^d$ where $\lambda = (p_1, \dots, p_d | q_1, \dots, q_d)$ in Frobenius notation. \end{proposition} A {\bf rim hook} is a connected set of boxes on the border of a Young diagram that does not contain a $2 \times 2$ square. \begin{proposition} For any $\mu \in \bY$ and $r \ge 1$, \[ p_r s_\mu = \sum_\lambda (-1)^{{\rm ht}(\lambda \setminus \mu)} s_\lambda \] where the sum is over all partitions $\lambda$ such that $\lambda \setminus \mu$ is a rim hook with $r$ boxes, and ${\rm ht}$ is the number of rows of the rim hook minus $1$. \end{proposition} \begin{proof} We have \[ p_r(x_1, \dots, x_N) s_\mu(x_1, \dots, x_N) = a_\delta(x)^{-1} \sum_j x_j^r \sum_{\sigma \in S_N} \sgn(\sigma) x_{\sigma(1)}^{\mu_1 + N - 1} \cdots x_{\sigma(N)}^{\mu_N} = a_\delta(x)^{-1} \sum_{k=1}^N a_{\mu + \delta + r\ol{e}_k}, \] where $\ol{e}_k$ is the $k$th standard basis vector. A basic argument shows that each $a_{\mu + \delta + r\ol{e}_k}$ is either 0 or can be rearranged to $(-1)^{{\rm ht}(\lambda \setminus \mu)} a_{\lambda + \delta}$ where $\lambda \setminus \mu$ is a rim hook. \end{proof} \begin{corollary}[Murnaghan--Nakayama rule] The coefficient of $s_\lambda$ in the decomposition of $p_\rho = p_{\rho_1} p_{\rho_2} \cdots$ is equal to \[ \sum_{\ul{\lambda}} (-1)^{{\rm ht}(S)} s_\lambda \] where the sum is over all chains $\ul{\lambda} : \emptyset \subset \lambda^{(1)} \subset \lambda^{(2)} \subset \cdots \subset \lambda^{\ell(\rho)} = \lambda$ such that $\lambda^{(i)} \setminus \lambda^{(i-1)}$ is a rim hook with $\rho_i$ boxes and ${\rm ht}(\ul{\lambda}) = \sum_{i=1}^{\ell(\rho)} {\rm ht}(\lambda^{(i)} \setminus \lambda^{(i-1)})$. \end{corollary} \subsubsection{Characters of the symmetric group, revisited} Consider a map $\psi \colon S_n \to \Lambda^n$ defined by $\psi(\sigma) = p_{\rho(\sigma)}$ where $\rho(\sigma)$ is the partition encoding the cycle lengths of $\sigma$. For any $f \in \bC[S_n]$, define the {\bf characteristic map} \[ {\rm ch}(f) = \frac{1}{n!} \sum_{\sigma \in S_n} f(\sigma) \psi(\sigma). \] \begin{theorem} ${\rm ch}(\chi^\lambda) = s_\lambda$ where $\chi^\lambda$ is the trace of the corresponding irreducible representation of $S_n$. \end{theorem} Let $c_\rho$ be the conjugacy class of type $\rho$. In particular, this says that \[ s_\lambda = {\rm ch}(\chi^\lambda) = \frac{1}{n!} \sum_{|\rho| = n} |c_\rho| \chi^\lambda_\rho p_\rho. \] Using the orthogonality relations, we get the Frobenius formula \[ p_\rho = \sum_{|\lambda|=n} \chi^\lambda_\rho s_\lambda. \] (This is currently the most efficient way to calculate the character values for the symmetric group.) Combining this with the Murnaghan--Nakayama rule, we get a combinatorial formula for $\chi^\lambda_\rho$ in terms of rim hooks. We define a product on characters of all symmetric groups via induction: $\chi^{\pi_1} \circ \chi^{\pi_2} = \chi^\pi$ where $\pi = \ind_{S_{n_1} \times S_{n_2}}^{S_{n_1+n_2}} \pi_1 \boxtimes \pi_2$. Under the characteristic map, this product goes to usual product of symmetric functions. \section{$S_\infty$} Define $S_\infty$ to be the group of permutations of $\{1, 2, 3, \dots \}$ that fixes all but finitely many elements. We can also write this as the direct limit of the tower of inclusions $S_1 \subset S_2 \subset S_3 \subset \cdots$. We wish to classify characters of $S_\infty$. Let $\chi \colon S_\infty \to \bC$ be a character. Then the restriction $\chi|_{S_n}$ is a character for $S_n$. But we know that $\chi|_{S_n} = \sum_{|\lambda|=n} M_n(\lambda) \frac{\chi^\lambda}{\dim \lambda}$ where $M_n(\lambda) \ge 0$ and $\sum_\lambda M_n(\lambda) = 1$. Now we have the relation \[ \sum_{|\mu|=n-1} M_{n-1}(\mu) \frac{\chi^\mu}{\dim \mu} = \chi|_{S_{n-1}} = \sum_{|\lambda| = n} \frac{M_n(\lambda)}{\dim \lambda} \chi^\lambda|_{S_{n-1}} = \sum_{|\lambda|=n} \frac{M_n(\lambda)}{\dim \lambda} \sum_{\nu \nearrow \lambda} \chi^\nu. \] In particular, we have \[ \frac{M_{n-1}(\mu)}{\dim \mu} = \sum_{\nu \nearrow \lambda} \frac{M_n(\lambda)}{\dim \lambda}. \] Now define $\phi(\lambda) = M_{|\lambda|}(\lambda) / \dim \lambda$. Then we have the relation $\phi(\mu) = \sum_{\mu \nearrow \lambda} \phi(\lambda)$, and $\phi(\mu) \ge 0$ and $\phi(1) = 1$. We will call such a $\phi$ a {\bf harmonic function}. Then the harmonic functions are naturally in bijection with characters of $S_\infty$. We also set $q_n(\lambda,\mu) = \dim \mu / \dim \lambda$ if $\mu \nearrow \lambda$ and 0 otherwise. We can write \[ M_{n-1}(\mu) = \sum_{\mu \nearrow \lambda} q_n(\lambda,\mu) M_n(\lambda). \] Now $M_n$ is a probability measure on $\bY_n$, and we can think of $q_n(\lambda,\mu)$ as stochastic projections. Hence harmonic functions are the same as systems of measures $\{M_n\}_n$ consistent with the stochastic projections $q_n$. Now we will define a probability measure on paths $\lambda^{(1)} \nearrow \lambda^{(2)} \nearrow \cdots$ with $|\lambda^{(j)}| = j$. The probability of all paths with $\lambda^{(n)}$ fixed is $\phi(\lambda^{(n)}) = M_n(\lambda^{(n)}) / \dim \lambda^{(n)}$. The correctness of this definition is equivalent to the harmonicity of $\phi$, and also to the coherency of the $M_n$ with respect to the $q_n$. %Lecture 4: 9/20/11 \begin{theorem}[Choquet] Assume that $X$ is a metrizable compact convex set in a locally convex topological linear space $V$ and pick $x_0 \in X$. Then there exists a (Borel) probability measure $P$ on $X$ supported by the set $E(X)$ of extreme points of $X$ which represents $x_0$, i.e., for any continuous linear functional $f \in V^*$, we have $f(x_0) = \int_{E(X)} f(x) P({\rm d}x)$. \end{theorem} reference: book by Phelps We can apply this theorem to the case when $V$ is the space of functions on Young diagrams and $X$ is the subset of harmonic functions. \begin{theorem}[Vershik--Kerov ``ring theorem''] If a nonnegative normalized harmonic function $\phi$ on $\bY$ is extreme, then the linear map $\hat{\phi} \colon \Lambda \to \bR$ defined by $\hat{\phi}(s_\lambda) = \phi(\lambda)$ is multiplicative, i.e., $\hat{\phi}(fg) = \hat{\phi}(f) \hat{\phi}(g)$. Furthermore, this correspondence is a bijection between extreme nonnegative normalized harmonic functions and algebra homomorphisms $F \colon \Lambda \to \bR$ such that $F(s_1) = 1$ and $F(s_\lambda) \ge 0$ for all $\lambda \in \bY$. \end{theorem} \begin{proof} First we show that $\phi$ extreme implies that $\hat{\phi}$ is multiplicative. For any $\lambda \in \bY$ such that $\phi(\lambda) \ne 0$, define $\phi_\lambda \colon \bY \to \bR$ by $\phi_\lambda(\mu) = \hat{\phi}(s_\lambda s_\mu) / \hat{\phi}(s_\lambda)$. We claim that $\phi_\lambda$ is harmonic. We have \begin{align} \label{eqn:harmonicphihat} \sum_{\nu: \nu \searrow \mu} \phi_\lambda(\nu) = \sum_{\nu \searrow \mu} \frac{\hat{\phi}(s_\lambda s_\nu)}{\hat{\phi}(s_\lambda)} = \frac{\hat{\phi}(s_\lambda \sum_{\nu \searrow \mu} s_\nu)}{\hat{\phi}(s_\lambda)} = \frac{\hat{\phi}(s_\lambda p_1 s_\mu)}{\hat{\phi}(s_\lambda)}. \end{align} Since $\phi$ is harmonic, we have \[ \hat{\phi}(s_\kappa) = \phi(\kappa) = \sum_{\rho \searrow \kappa} \phi(\rho) = \hat{\phi}(\sum_{\rho \searrow \kappa} s_\rho) = \hat{\phi}(p_1s_\kappa). \] This implies that $\hat{\phi}(p_1f) = \hat{\phi}(f)$ for any $f \in \Lambda$. Going back to \eqref{eqn:harmonicphihat}, we get that the result is equal to $\hat{\phi}(s_\mu s_\lambda) / \hat{\phi}(s_\lambda) = \phi_\lambda(\mu)$. Also, from Schur-positivity of products of Schur functions, $\phi_\lambda$ is nonnegative. This proves the claim. Similarly, for any $f = \sum_\mu c_\mu s_\mu$ with $c_\mu \ge 0$ and $\hat{\phi}(f) \ne 0$, we define $\phi_f(\mu) = \hat{\phi}(fs_\mu) / \hat{\phi}(f)$. The argument above shows that $\phi_f$ is harmonic, normalized, and nonnegative. Now we claim that that $\hat{\phi}(s_\mu s_\lambda) = \hat{\phi}(s_\mu) \hat{\phi}(s_\lambda)$. If $\phi(\lambda) \hat{\phi}(s_\lambda) = 0$, then $\phi(\rho) = 0$ for any $\rho$ that contains $\lambda$ from the harmonicity condition. Hence the claim holds in this case. So assume that $\hat{\phi}(s_\lambda) \ne 0$ and set $n=|\lambda|$. Take $f_1 = s_\lambda / 2$, $f_2 = p_1^n - f_1 = \sum_{|\mu| = n} (\dim \mu) s_\mu - s_\mu / 2$. So $f_2$ is Schur-positive. Then we have \[ \phi = \hat{\phi}(f_1) \phi_{f_1} + \hat{\phi}(f_2) \phi_{f_2}. \] To see this, pick $\mu$. Then the left-hand side is \[ \phi(s_\mu) = \hat{\phi}(f_1) \frac{\hat{\phi}(f_1s_\mu)}{\hat{\phi}(f_1)} + \hat{\phi}(f_2) \frac{\hat{\phi}(f_2 s_\mu)}{\hat{\phi}(f_2)} = \hat{\phi}((f_1 + f_2)s_\mu) = \hat{\phi}(p_1^n s_\mu) = \hat{\phi}(s_\mu). \] Since $\phi$ was assumed to be extreme, we must have either $\phi = \phi_{f_1}$ or $\phi = \phi_{f_2}$. In the first case, we get the identity $\hat{\phi}(s_\lambda s_\nu / 2) / \hat{\phi}(s_\lambda / 2) = \hat{\phi}(s_\nu)$ which proves multiplicativity. \steven{other case similar?} Now we will show the reverse direction. Let $H_\bY$ be the set of all nonnegative normalized harmonic functions on $\bY$, which lives in the space of all functions on $\bY$, with the topology of pointwise convergence. By Choquet's theorem, for any $\phi$, we have $\phi(y) = \int_{E(H_\bY)} \psi(\lambda) P({\rm d}\psi)$ for some probability measure $P$. This is equivalent to saying $\hat{\phi}(s_\lambda) = \int_{E(H_\bY)} \hat{\psi}(s_\lambda) P({\rm d}\psi)$. By the first part of the proof, each $\hat{\psi}$ is multiplicative. Now assume that $\phi$ is multiplicative. Then we have \[ \left( \int_{E(H_\bY)} \hat{\psi}(f) P({\rm d}\psi)\right)^2 = \hat{\phi}(f)^2 = \hat{\phi}(f^2) = \int_{E(H_\bY)} \hat{\psi}(f^2) P({\rm d}\psi) = \int_{E(H_\bY)} (\hat{\psi}(f))^2 P({\rm d}\psi). \] If we think of $\hat{\psi}(f)$ as a random variable, this equality says ${\rm Var}(\hat{\psi}(f)) = 0$. This implies that $\hat{\psi}(f)$ is a constant with probability 1. Now take $f = s_\lambda$ and take a set $A_\lambda \subset E(H_\bY)$ such that $\psi \in A_\lambda$ if $\psi(s_\lambda) = \phi(s_\lambda)$. We know that $P(A_\lambda) = 1$. Then $\bigcap_\lambda A_\lambda \ne \emptyset$. So for any $\psi \in \bigcap_\lambda A_\lambda$, we will have $\psi = \phi$ which implies that $\phi$ is extreme. \end{proof} Vague analogy: positive-definite normalized continuous functions on $\bR$ are $\psi \colon \bR \to \bC$ such that $\psi(0) = 1$ and $[\psi(x_i - y_j)] \ge 0$ when $x_1 > x_2 > \cdots $ and $y_1 > y_2 > \cdots$. \begin{theorem}[Bochner] Such $\psi$ are in bijection with probability measures $\mu$ on $\bR$: $\psi(x) = \int_\bR \exp(ixp) \mu({\rm d}p)$. \end{theorem} Note that $\exp(ixp) \colon \bR \to \bC$ are the characters of $\bR$. \subsection{Consequences of Vershik--Kerov ``ring theorem''} Let $\chi$ be an extreme character for $S_\infty$ and let $\rho$ be a partition. Then $\chi(\rho) = \chi|_{S_n}(\rho)$ for some $n = |\rho|$. Write this as \[ \sum_{|\lambda| = n} \phi(\lambda) \chi^\lambda(\rho) = \sum_{|\lambda| = n} \hat{\phi}(s_\lambda) \chi^\lambda(\rho) = \hat{\phi}(\sum_{|\lambda| = n} \chi^\lambda(\rho) s_\lambda) = \hat{\phi}(p_\rho) = \hat{\phi}(p_{\rho_1}) \hat{\phi}(p_{\rho_2}) \cdots = \chi(\rho_1) \chi(\rho_2) \cdots. \] This is a simplification over the finite-dimensional case. The conjugacy classes for $S_\infty$ form a semigroup, where the addition is given by union of partitions. The above calculation shows that extreme characters respect this semigroup structure. (This is typical.) ~ What is the condition $F(s_\lambda) \ge 0$? Recall Jacobi--Trudi: $s_\lambda = \det(h_{\lambda_i - i + j})_{i,j=1}^N$ for any $N \ge \ell(\lambda)$. Applying multiplicative $F$ to both sides of this identity, we get $F(s_\lambda) = \det(F(h_{\lambda_i - i + j}))_{i,j=1}^N = \det(c_{\lambda_i - i + j})_{i,j=1}^N$ where $c_n = F(h_n) \ge 0$. Now consider the infinite matrix \[ (c_{i-j})_{i,j \ge 0} = \begin{pmatrix} 1 & c_1 & c_2 & c_3 & c_4 \\ 0 & 1 & c_1 & c_2 & c_3 \\ 0 & 0 & 1 & c_1 & c_2 & \cdots\\ \vdots \end{pmatrix} \] Then the Jacobi--Trudi identity is equivalent to saying that all minors (with consecutive indices) of this infinite matrix are nonnegative. By some formal argument, we can remove the restriction on indices being consecutive. We say that the matrix is {\bf totally positive} in this case. The problem of characterizing such matrices was solved by Edrei (1953) and the problem of characterizing extreme characters of $S_\infty$ was solved by Thoma (1964). The connection was not noticed until much later. \begin{theorem}[Edrei--Thoma] Let $F \colon \Lambda \to \bR$ be an algebra homomorphism with $F(p_1) = 1$. Then the following are equivalent: \begin{compactenum} \item $F(s_\lambda) \ge 0$ for all $\lambda \in \bY$. \item There exists a pair of nonnegative weakly decreasing sequences $\alpha = (\alpha_1 \ge \alpha_2 \ge \cdots \ge 0)$, $\beta = (\beta_1 \ge \beta_2 \ge \cdots \ge 0)$ such that $\sum_{j \ge 1} (\alpha_j + \beta_j) \le 1$ and such that the values of $F$ on power sums are given by $F(p_m) = \sum_{j \ge 1} \alpha_j^m - \sum_{j \ge 1} (-\beta_j)^m$ for $m \ge 2$. \end{compactenum} Furthermore, the sequences $\{\alpha_j\}$, $\{\beta_j\}$ are uniquely determined by $F$. \end{theorem} The character coming from such an $(\alpha, \beta)$ is given by \[ \chi^{(\alpha, \beta)}(\rho) = \prod_{m \ge 1} (\sum_j \alpha_j^{\rho_m} - \sum_j (-\beta_j)^{\rho_m}). \] Note that \[ F^{(\alpha, \beta)}(H(t)) = 1 + \sum_{n \ge 1} F(h_n) t^n = e^{\gamma t} \prod_{j \ge 1} \frac{1 + \beta_jt}{1-\alpha_jt}, \] where $\gamma = 1 - \sum_{j \ge 1} (\alpha_j + \beta_j) \ge 0$. This set of functions is closed under multiplication and renormalization of parameters. This is related to the fact that if $A$ and $B$ are totally positive matrices, then $AB$ is also totally positive by the identity $(AB)_{i_1,\dots,i_m}^{j_1,\dots,j_m} = \sum_{k_1 < \cdots < k_m} A_{i_1, \dots, i_m}^{k_1, \dots, k_m} B_{k_1, \dots, k_m}^{j_1, \dots, j_m}$. We can think of $(1-\alpha t)^{-1}$, $(1+\beta t)$ and $e^{\gamma t}$ as the building blocks of all extreme characters. \begin{definition} Let $\{\lambda(n)\}_{n \ge 1}$ be a sequence of Young diagrams with $|\lambda(n)| = n$. This sequence is {\bf regular} if for any $\sigma \in S_\infty$, the limit \[ \lim_{n \to \infty} \frac{\chi^{\lambda(n)}(\sigma)}{\dim \lambda(n)} \] exists. \end{definition} \begin{theorem} $\{\lambda(n)\}_{n \ge 1}$ is regular if and only if the limits \[ \alpha_i := \lim_{n \to \infty} \frac{\lambda(n)_i}{n}, \quad \beta_i := \lim_{n \to \infty} \frac{\lambda(n)'_i}{n} \] exist. In this case, we have \[ \lim_{n \to \infty} \frac{\chi^{\lambda(n)}(\sigma)}{\dim \lambda(n)} = \chi^{(\alpha, \beta)}(\sigma). \] \end{theorem} \subsection{Toy example: Pascal graph} Set $\cP_n = \{(k, \ell) \in \bZ^2_{\ge 0} \mid k+\ell = n\}$. The problem is to find all (extreme) nonnegative normalized harmonic functions on $\cP$. This is equivalent to finding all functions $\phi \colon \bZ^2_{\ge 0} \to [0,1]$ such that $\phi(k,\ell) = \phi(k+1,\ell) + \phi(k,\ell+1)$. Note that $\cP \subset \bY$ by identifying $(k,\ell)$ with the hook partition $(k+1, 1^{\ell})$. \begin{theorem} \label{thm:pascalharmonic} The nonnegative normalized harmonic functions on $\cP$ are in bijection with probability measures on $[0,1]$: given a probability measure $\mu$, the corresponding harmonic function is \[ \phi(k,\ell) = \int_0^1 p^k(1-p)^\ell \mu({\rm d}p). \] \end{theorem} Consider the space $\{0,1\}^{\N} = \{(w_1, w_2, \dots) \mid w_i \{0,1\}\}$. We are interested measures on this space that are invariant with respect to the natural $S_\infty$-action. In this case, extreme points are called {\bf ergodic measures}. With this terminology, we can rephrase the above theorem as follows. \begin{theorem} The $S_\infty$-invariant probability measures on $\{0,1\}^{\N}$ are in bijection with probability measures on $[0,1]$: given a probability measure $\mu$ on $[0,1]$, the corresponding $S_\infty$-invariant measure on $\{0,1\}^\N$ is \[ \nu = \int_0^1 \nu_p^{\otimes \infty} \mu({\rm d}p), \] where $\nu_p^{\otimes \infty} = \nu_p \otimes \nu_p \otimes \cdots$ and $\nu_p$ is the Bernoulli measure on $\{0,1\}$ given by $\nu_p(0) = p$ and $\nu_p(1) = 1-p$. \end{theorem} \begin{remark} This is a special case of de Finetti's theorem that deals with $S_\infty$-invariant measures on $X^{\N}$ where $X$ is any space. \end{remark} Why are these statements equivalent? Assume we have $\nu$, which is an $S_\infty$-invariant measure on $\{0,1\}^\N$. Then define \begin{align*} \phi(k,\ell) &= \nu(\{w_\bullet \mid w_1 = \cdots = w_k = 0, \ w_{k+1} = \cdots = w_{k+\ell} = 1 \}). \end{align*} Then $\phi(k,\ell) = \phi(k+1,\ell) + \phi(k,\ell+1)$ by the $S_\infty$-invariance of $\nu$. %Disclaimer: in what follows, we will not mention caveats about sets of %measure 0. An ergodic measure $\nu$ is defined by the condition that any $S_\infty$-invariant subset in $\{0,1\}^\N$ has either measure 0 or 1. Why is ergodic the same as extreme? First suppose that $\nu$ is not ergodic. Then there is some $S_\infty$-invariant subset $S$ whose measure is neither 0 nor 1. Then we can consider two measures $\nu|_S$ and $\nu|_{\{0,1\}^\N \setminus S}$. After renormalizing, we can write $\nu$ as the sum of these two measures which shows that it is not extreme. Now suppose that $\nu$ is not extreme. So in the theorem above, $\mu$ is not a $\delta$-measure on $[0,1]$. Then we can define intervals $I_1, I_2 \subset [0,1]$ such that $\mu(I_1) \ne 0$ and $\mu(I_2) \ne 0$. Let $p_0$ be the distance between $I_1$ and $I_2$. Now define $S$ to be a set of sequences $(\eps_1, \eps_2, \dots)$ such that \[ \lim_{n \to \infty} \frac{\#\{1\text{'s in } (\eps_1, \dots, \eps_n)}{n} > p_0. \] This set is $S_\infty$-invariant, which shows that $\nu$ is not ergodic. \begin{proof}[Proof 1 of Theorem~\ref{thm:pascalharmonic}] Extreme $\phi$ are in bijection with algebra homomorphisms $F \colon \bR[x,y] \to \bR$ such that $F(x+y) = 1$ and $F(x^ky^\ell) \ge 0$ for all $k,\ell$. Motivation: recall from before that we considered algebra homomorphisms $F \colon \Lambda \to \bR$ such that $F(s_\mu) \ge 0$ and $F(p_1) = 1$. Also multiplication by $p_1$ encodes the branching structure of the graph. So proof of the above statement follows in the same way. So given such an $F$, we have $F(x^ky^\ell) = F(x)^k F(y)^\ell = F(x)^k (1-F(x))^\ell$. If we set $F(x) = p$, then we are done: a probability measure $\mu$ on $[0,1]$ is uniquely determined by its moments $m_k = \int_0^1 x^k \mu({\rm d}x)$ since polynomials are dense in $C[0,1]$, and $C[0,1]^*$ is the space of signed measures on $[0,1]$. \end{proof} We will give another proof that will generalize to the Young graph after some preparatory lemmas. \begin{definition} $\dim(k,\ell) = \binom{k+\ell}{k}$ and $\dim((k_0, \ell_0), (k,\ell)) = \dim(k-k_0, \ell-\ell_0) = \binom{k+\ell-k_0-\ell_0}{k-k_0}$. \end{definition} \begin{lemma} \label{lemma:pascalasymptotic} \[ \frac{\dim((k_0,\ell_0), (k,\ell))}{\dim(k,\ell)} = \left(\frac{k}{k+\ell}\right)^{k_0} \left(\frac{\ell}{k+\ell}\right)^{\ell_0} + O\left(\frac{1}{k+\ell} \right). \] \end{lemma} \begin{proof} Denote $a^{\downarrow k} = a(a-1) \cdots (a-k+1)$. The left-hand side is \[ \frac{k^{\downarrow k_0} \ell^{\downarrow \ell_0}}{(k+\ell)^{\downarrow (k_0 + \ell_0)}} = \frac{(k^{k_0} + \cdots)(\ell^{\ell_0} + \cdots)}{(k+\ell)^{k_0 + \ell_0} + \cdots}, \] which is the right-hand side. \end{proof} \begin{lemma} \label{lemma:convergentprobability} The set of probability measures on a compact set is weakly compact (i.e., any sequence of probability measures on a compact set has a weakly convergent subsequence to a probability measure). \end{lemma} \begin{proof} In general, the unit ball in the dual space to a normed space is compact in $w^*$-topology. For a compact $K$, take the normed space to be $C(K)$ with the norm $\|f\| = \max|f|$. Then the dual space is the space of signed measures. \end{proof} \begin{proof}[Proof 2 of Theorem~\ref{thm:pascalharmonic}] We have $\phi(k,\ell) = \Prob(\{(0^k,1^\ell, *, *, \dots)\})$. For $n=k+\ell$, we define \[ M_n(k,\ell) = \phi(k,\ell) \binom{n}{k} = \Prob \{ \text{exactly $k$ zeroes in the first $n$ coordinates}\}. \] Note that $\sum_{k+\ell=n} M_n(k,\ell) = 1$. Note that we have the identity \[ M_n(k,\ell) = \frac{\binom{n}{k}}{\binom{n+1}{k}} M_{n+1}(k,\ell+1) + \frac{\binom{n}{k}}{\binom{n+1}{k+1}} M_{n+1}(k+1,\ell). \] We write \begin{align*} \phi(k_0, \ell_0) &= \sum_{k+\ell=n} \dim((k_0, \ell_0), (k,\ell)) \phi(k, \ell) = \sum_{k+\ell=n} \frac{\dim((k_0,\ell_0), (k,\ell))}{\dim(k, \ell)} M_n(k,\ell)\\ &= \sum_{k+\ell=n} \left( \left( \frac{k}{k+\ell} \right)^{k_0} \left( \frac{ \ell}{k+\ell} \right)^{\ell_0} + O\left( \frac{1}{n} \right) \right) M_n(k,\ell)\\ &= \sum_{k+\ell=n} \left( \frac{k}{k+\ell} \right)^{k_0} \left( \frac{ \ell}{k+\ell} \right)^{\ell_0} M_n(k,\ell) + O\left( \frac{1}{n} \right) \end{align*} using Lemma~\ref{lemma:pascalasymptotic}. For any $p \in [0,1]$, denote by $\langle p \rangle$ the $\delta$-measure at $p$. Start with a harmonic function $\phi$. Consider the sequence of probability measures on $[0,1]$ given by \[ M^{(n)} = \sum_{k+\ell = n} M_n(k,\ell) \langle \frac{k}{n} \rangle. \] By Lemma~\ref{lemma:convergentprobability}, the sequence $\{M^{(n)}\}_n$ has a convergent subsequence. On the other hand, we have \[ \phi(k_0, \ell_0) = \int_0^1 p^{k_0}(1-p)^{\ell_0} M^{(n)} ({\rm d}p) + O\left(\frac{1}{n}\right). \] Since the left-hand side doesn't depend on $n$, we take $n \to \infty$ to get \[ \phi(k_0, \ell_0) = \int_0^1 p^{k_0}(1-p)^{\ell_0} \mu({\rm d}p) \] for some probability measure $\mu$ on $[0,1]$. \end{proof} \begin{remark} $[0,1]$ is the boundary of $\cP$. Intuitively, the levels of $\cP$ are limiting to $[0,1]$. As an extension, one can consider the graph of Young diagrams with at most $D$ diagonal boxes. \end{remark} %9/27/11 \begin{remark} A vague analogy: Define $D_1 = \{u \in \bC \mid |u| < 1\}$. Take $\phi \colon D_1 \to \bC$ be to be harmonic (i.e., $\Delta \phi = 0$), nonnegative ($\phi \ge 0$), and normalized so that $\phi(0) = 1$. then $\frac{1}{2\pi r} \phi |_{\{|u|=r\}}$ is the density of a probability measure on a circle. Taking $r \to 1$, these probability measures have a unique limit $\mu$, and we have the Poisson kernel \[ \phi(z) = \frac{1}{2\pi} \int_{|\zeta|=1} \frac{1 - |z|^2}{|z-\zeta|^2} \mu({\rm d}\theta) \] where $\zeta = \exp(i\theta)$ on $\partial D_1$. \end{remark} \subsection{Super symmetric functions} \begin{lemma} The super power sums of order $k \ge 2$ of $(\alpha, \beta)$ are \[ p^\circ_k(\alpha, \beta) = \sum_{i=1}^\infty \alpha_1^k - \sum_{i=1}^\infty (-\beta_i)^k. \] These functions are well-defined and continuous on $\Omega$. \end{lemma} \begin{proof} Since the $\alpha_i$ decrease and $\sum_i \alpha_i \le 1$, we have $\alpha_i \le 1/i$ for all $i \ge 1$. Similarly, $\beta_j \le 1/j$ for all $j$. Then $\sum_i \alpha_i^k$ and $\sum_j \beta_j^k$ are uniformly convergent on $\Omega$ which implies they belong to $C(\Omega)$. \end{proof} \begin{remark} The function $(\alpha, \beta) \mapsto \sum_{i=1}^\infty (\alpha_i + \beta_i)$ is not continuous on $\Omega$. So by convention, we will just define $p_1^\circ(\alpha, \beta) = 1$. \end{remark} We can define the map $\Lambda \to C(\Omega)$ by $p_j \mapsto p^\circ_j$. So for any $f \in \Lambda$, get $f^\circ \in C(\Omega)$. In particular, for each partition $\lambda \in \bY$, we can define $s^\circ_\lambda$. \begin{theorem}[Edrei--Thoma] \label{thm:edreithoma} The nonnegative normalized harmonic functions on $\bY$ are in bijective correspondence with (Borel) probability measures on \[ \Omega = \{ \alpha_1 \ge \alpha_2 \ge \cdots \ge 0,\ \beta_1 \ge \beta_2 \ge \cdots \ge 0 \mid \sum_{j=1}^\infty (\alpha_j + \beta_j) \le 1 \} \subset [0,1]^\infty \times [0,1]^\infty. \] Given a probability measure $P$, the corresponding harmonic function $\phi$ is given by \[ \phi(\lambda) = \int_{\omega \in \Omega} s^\circ_\lambda(\omega) P({\rm d}\omega). \] \end{theorem} \begin{theorem}[Stone--Weierstrass] Let $X$ be a compact topological space, let $C(X)$ be the algebra of real-valued continuous on $X$, equipped with sup-norm. Let $A \subset C(X)$ be a closed (with respect to the norm topology) subalgebra (containing the function $1$) that separates points (i.e., for all $x_1 \ne x_2 \in X$, there is $f \in A$ such that $f(x_1) \ne f(x_2)$). Then $A = C(X)$. \end{theorem} \begin{remark} The statement is wrong if one drops the assumption $1 \in A$. \end{remark} \begin{lemma} The functions $p_k^\circ$ for $k \ge 1$ separate points of $\Omega$. Consequently, the image of $\Lambda$ under $f \mapsto f^\circ$ is dense in $C(\Omega)$. \end{lemma} \begin{proof} Consider the function \[ \sum_{k=1}^\infty \frac{p^\circ_k(\alpha, \beta)}{u^k} = \sum_{i=1}^\infty \frac{\alpha_i}{u-\alpha_i} + \sum_{i=1}^\infty \frac{\beta_i}{u + \beta_i} + \frac{\gamma}{u} \] (recall that $\gamma = 1 - \sum_i (\alpha_i + \beta_i)$). This has poles at the 0, $\alpha_i$ and $\beta_j$, with the correct multiplicity. In particular, this function recovers $(\alpha, \beta)$ by looking at the poles / residues. \end{proof} \begin{corollary} Extreme nonnegative normalized harmonic functions on $\bY$ are in bijective correspondence with points of $\Omega$. In particular, given $\omega \in \Omega$, the corresponding function is \[ \phi(\lambda) = s_\lambda^\circ(\omega). \] \end{corollary} If $\lambda = (p_1, \dots, p_d | q_1, \dots, q_d)$ in Frobenius notation, then we introduce {\bf modified Frobenius notation} $\lambda = (p_1 + \frac{1}{2}, \dots p_d + \frac{1}{2} | q_1 + \frac{1}{2}, \dots, q_d + \frac{1}{2}) = (a_1, \dots, a_d | b_1, \dots, b_d)$. Now we define an embedding $\bY_n \subset \Omega$ via $\lambda \mapsto \omega_\lambda = (\alpha, \beta) \in \Omega$ where \[ \alpha = \left(\frac{a_1}{n}, \frac{a_2}{n}, \dots, 0, 0, \dots\right), \quad \beta = \left(\frac{b_1}{n}, \frac{b_2}{n}, \dots, 0, 0, \dots\right). \] Note that $\omega_\lambda \in \Omega_0 = \{(\alpha, \beta) \mid \sum_j (\alpha_j + \beta_j) = 1\}$. Note that $\Omega_0 \subset \Omega$ is dense despite initial appearances. (This is the statement in functional analysis that the closure of the unit sphere in an infinite-dimensional space is the unit-ball.) \begin{theorem} \label{thm:schurasymptotic} Let $\dim(\mu, \lambda)$ denote the number of paths in $\bY$ from $\mu$ to $\lambda$ obtained by adding 1 box at each step. Then for any fixed $\mu \in \bY$ and varying $\lambda \in \bY$, we have \[ \frac{\dim(\mu, \lambda)}{\dim(\lambda)} = s_\mu^\circ(\omega_\lambda) + O(|\lambda|^{-1/2}). \] \end{theorem} \begin{proposition} Let $\{\lambda^{(n)}\}_{n \ge 1}$ be a sequence of Young diagrams, where $|\lambda^{(n)}| \to \infty$ for $n \to \infty$. Then the following conditions are equivalent: \begin{compactenum}[\rm (1)] \item $\DS \lim_{n \to \infty} \frac{\dim(\mu, \lambda^{(n)})}{\dim(\lambda^{(n)})}$ exists for any $\mu$ (i.e., $\DS \lim_{n \to \infty} \frac{\chi^{\lambda^{(n)}}(\sigma)}{\dim(\lambda^{(n)})}$ exists) \item $\DS \lim_{n \to \infty} \omega_{\lambda^{(n)}}$ converges. \end{compactenum} \end{proposition} \begin{proof} Assume $\omega_{\lambda^{(n)}} \to \omega \in \Omega$. Then by Theorem~\ref{thm:schurasymptotic}, \[ \frac{\dim(\mu, \lambda^{(n)})}{\dim(\lambda^{(n)})} = s^\circ_\mu(\omega_{\lambda^{(n)}}) + O(|\lambda^{(n)}|^{-1/2}), \] which converges to $s_\mu^\circ(\omega)$ for $n \to \infty$. Conversely, assume that the limit in (1) exists. Take $\{ \omega_{\lambda^{(n)}} \} \subset \Omega$ which is compact. Then there exists a convergent subsequence $\omega_{\lambda^{(n_k)}} \to \omega_0$. This means that the limit in (1) must be $s^\circ_\mu(\omega_0)$. Since the $s^\circ_\mu$ separate points of $\Omega$, we see that the limit in (2) exists and is equal to $\omega_0$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:edreithoma}] Define $M^{(n)}(\lambda) = \dim(\lambda) \phi(\lambda)$ where $n = |\lambda|$, $\phi$ is harmonic. Then $M^{(n)}$ is a probability distribution on $\bY_n$. Let $\langle \omega \rangle$ be the delta-measure on $\Omega$ concentrated at $\omega \in \Omega$. Define \[ P^{(n)} = \sum_{|\lambda| = n} M^{(n)}(\lambda) \langle \omega_\lambda \rangle. \] This is a probability measure on $\Omega$ supported by $|\bY_n|$. By a general lemma, the sequence $\{P^{(n)}\}$ has a limit point $P$. Then \begin{align*} \phi(\mu) &= \sum_{\lambda \searrow \mu} \phi(\lambda) = \cdots = \sum_{\lambda \in \bY_n} \dim(\mu, \lambda) \phi(\lambda) = \sum_{\lambda \in \bY_n} \frac{\dim(\mu, \lambda)}{\dim(\lambda)} M^{(n)}(\lambda)\\ &= \sum_{\lambda \in \bY_n} (s^\circ_\mu(\omega_\lambda) + O(n^{-1/2})) M^{(n)}(\lambda) = \sum_{\lambda \in \bY_n} s_\mu^\circ (\omega_\lambda) M^{(n)}(\lambda) + O(n^{-1/2})\\ &= \int_{\omega \in \Omega} s^\circ_\mu(\omega) P^{(n)}({\rm d}\omega) + O(n^{-1/2}), \end{align*} which converges to $\int_\Omega s^\circ_\mu(\omega) P({\rm d}\omega)$. So we have defined a probability measure $P$ associated to $\phi$. It remains to observe that for any probability measure $P$, the function \[ \mu \mapsto \int_\Omega s^\circ_\mu(\omega) P({\rm d}\omega) \] is a normalized nonnegative harmonic function. Nonnegativity follows from the previous part since $s^\circ_\mu$ is a limit of nonnegative quantities. The fact that it is normalized is by the convention $s_1^\circ = p_1^\circ = 1$. For harmonicity, we note that \[ s^\circ_\mu = p_1^\circ s_\mu^\circ = (p_1 s_\mu)^\circ = \sum_{\nu \searrow \mu} s_\nu^\circ. \qedhere \] \end{proof} Note that we have skipped the asymptotics of $\dim(\mu, \lambda) / \dim(\lambda)$, which is the hard part. \begin{lemma} \label{lemdimskew} For any two partitions $\mu, \lambda$, and any $N \ge \ell(\lambda)$, we have \[ \frac{\dim(\mu, \lambda)}{(|\lambda| - |\mu|)!} = \det\left( \frac{1}{(\lambda_i - i - (\mu_j - j))!} \right)_{i,j=1}^N \] with the convention that the inverse of a negative factorials is $0$. \end{lemma} \begin{remark} For $\mu = \emptyset$, this is the hook formula. \end{remark} \begin{proof} If $|\lambda| < |\mu|$, then the formula gives 0, and if $|\lambda| = |\mu|$, then both sides of the formula gives $\delta_{\lambda, \mu}$. So we will write $|\lambda| = n > m = |\mu|$ and work with $n$ variables. Note that \[ p_1^{n-m} \frac{a_{\mu + \delta}}{a_\delta} = p_1^{n-m} s_\mu = \sum_{\nu \in \bY_n} \dim(\mu, \nu) s_\nu = \sum_{\nu \in \bY_n} \dim(\mu, \nu) \frac{a_{\nu+\delta}}{a_\delta}. \] So we have \[ p_1^{n-m} a_{\mu + \delta} = \sum_{\nu \in \bY_n} \dim(\mu, \nu) a_{\nu + \delta}. \] Now we compare the coefficients of $x_1^{\nu_1 + n - 1} \cdots x_n^{\nu_n}$ of both sides. We have \[ p_1^{n-m} = (x_1 + \cdots + x_n)^{n-m} = \sum_{k_1 + \cdots + k_n = n-m} \frac{(n-m)!}{k_1! \cdots k_n!} x_1^{k_1} \cdots x_n^{k_n}, \] and \[ a_{\mu + \delta} = \sum_{\sigma \in S_n} \sgn(\sigma) x_1^{\mu_{\sigma(1)} + n - \sigma(1)} \cdots x_n^{\mu_{\sigma(n)}}, \] and the formula follows from this. \end{proof} Idea for the asymptotics: \[ \frac{\dim(\mu, \lambda)}{\dim(\lambda)} = \frac{m!}{n!} s_\mu^*(\lambda) \] where $s^*_\mu$ is a {\bf shifted Schur function}, and we will show that it approximates $s^\circ_\mu(\omega_\lambda)$ by identifying the algebra generated by the shifted Schur functions and working with a set of generators. %September 29 The key estimate \[ \frac{\dim(\mu,\lambda)}{\dim \lambda} = s^{\circ}_{\mu}(\lambda) + O\left(\frac{1}{|\lambda|^{1/2}} \right) \] where $\dim(\mu,\lambda)=\#\{\text{paths in $\bY$ from $\lambda$ to $\mu$}\}$, $f\mapsto f^{\circ}$ is an embedding $\Lambda \hookrightarrow \mathcal{C}(\Omega)$ where $\Omega = \{\alpha_1\geq \alpha_2 \geq \cdots, \beta_1 \geq \beta_2 \geq \cdots \mid \sum_j (\alpha_j + \beta_j) \leq 1\}$ whose image on Newton power sums is \[ p^{\circ}_k(\omega) = \begin{cases} 1 & k=1\\ \sum (\alpha_j)^k - \sum(-\beta_j)^k , & k\geq 2 \end{cases} \] and $\omega_{\lambda} \in \Omega$, $\omega_{\lambda} = \left(\frac{a_1}{|\lambda|}, \frac{a_2}{|\lambda|},\ldots \mid \frac{b_1}{|\lambda|}, \frac{b_2}{|\lambda|},\ldots\right)$. \begin{definition} The {\bf shifted Schur polynomial} in $N$ variables is \[ s^{*}_{\mu}(x_1,\ldots,x_N) = \frac{\det[(x_i+N-i)^{ \downarrow(\mu_j+N-j)}]}{\det[(x_i+N-i)^{\downarrow(N-j)}]_{i,j=1}^N} \] for $\mu \in \bY$, $\ell(\mu)\leq N$ (otherwise $s^*_{\mu}=0$) and where $x^{\downarrow k}$ is the falling factorial: \[ x^{\downarrow k} = \begin{cases} x(x-1)\cdots (x-k+1) & k\geq 1\\ 1 & k=0 \end{cases} \qedhere \] \end{definition} (Recall the definition of the Schur polynomial $s_{\mu}(x_1,\ldots,x_N) =\frac{\det[x_i^{\mu_i+N-j}]}{\det[x_i^{N-j}]}$.) The polynomial $s^*_{\mu}$ is symmetric in the variables $\{x_i-i\}_{i=1}^N$. Also, the denominator is the (shifted) Vandermonde $\prod_{1\leq i i \}. \] If $X = \varprojlim V_i$ exists, then there is a map $\cM(X) \to \cM_\infty$. For any $\mu \in \cM(X)$, $\mu_i := \mu \Lambda_i^\infty$ is a coherent system. \subsection{Projective chains.} The index set is $\{1, 2, \dots\}$, the links are $\{\Lambda_N^{N+1} \mid N \ge 1\}$. We also assume that the $V_N$ are discrete (finite or countable). In this case, the links are matrices $[\Lambda_N^{N+1}(x,y)]_{x \in V_{N+1}, y \in V_N}$ with the property that $\sum_{y \in V_N} \Lambda_N^{N+1}(x,y) =1$ for any $x \in V_{N+1}$ and $\Lambda_N^{N+1}(x,y) \ge 0$ for all $x,y$. For any $N' > N$, the composition $\Lambda_N^{N'} = \Lambda^{N'}_{N'-1} \cdots \Lambda^{N+1}_N$ is given by matrix product. A coherent system of measures $\{M_n(x)\}_{x \in V_n}$ is defined by the properties $\sum_{x \in V_n} M_n(x) = 1$ and $\sum_{x \in V_{N+1}} M_{N+1}(x) \Lambda_N^{N+1}(x,y) = M_N(y)$ for all $y \in V_N$. \begin{example} The set $\cM_\infty$ may be empty. For example, take $V_n =\{n,n+1,n+2,\dots\}$, and $\Lambda_N^{N+1}$ is the natural embedding $\{N+1, N+2, \dots\}$ into $\{N,N+1,N+2,\dots\}$. \end{example} \begin{proposition} If all $V_N$ are finite, then $\cM_\infty \ne \emptyset$. \end{proposition} We will assume from now on that $\cM_\infty \ne \emptyset$. One can view $\cM_\infty$ as a subset in $\bR^{\coprod_{i \ge 1} V_i}$ just by assigning to a coherent system the set of values it takes on each singleton. The latter space is a metrizable locally convex space. So $\cM_\infty$ is a convex subset, which implies that $\cM_\infty$ is a standard Borel space. Let $V_\infty$ be the set of extreme points in $\cM_\infty$. Call $V_\infty$ the {\bf boundary} of $\{V_i, \Lambda_i^j\}$. \begin{theorem} If $\cM_\infty$ is nonempty, then the boundary $V_\infty \subset\cM_\infty$ is a nonempty measurable subset of $\cM_\infty$ (in fact, a $G_\delta$-set), and there is a natural isomorphism of measurable spaces $\cM_\infty \cong \cM(V_\infty)$. \end{theorem} This can be derived from Choquet's theorem. $V_\infty$ comes with links $\Lambda_i^\infty \colon V_\infty \to V_i$. To define these, note that $V_\infty \subset \cM_\infty$. So each $v_\infty \in V_\infty$ defines a coherent system of measures $\{M_N\}_{N \ge 1}$ by \[ \Lambda_N^\infty(v_\infty,x) = M_N(x). \] The identities $\Lambda_N^\infty \Lambda_M^N = \Lambda_M^\infty$ follows from the coherence of $\{M_N\}_{N \ge 1}$. One can show that this gives an identification $V_\infty = \varprojlim V_i$. \subsection{Graded graphs.} \begin{definition} A {\bf graded graph} $\Gamma$ is a graph (multiedges allowed) with countably many vertices, partitioned into levels $1, 2, 3, \dots$ (for a vertex $v$, let $|v|$ denote its level) with the following conditions: \begin{compactenum} \item For any two vertices $v,v'$ which are joined by an edge, we have $|v| = |v'| \pm 1$. \item Each vertex of level $i$ is joined with at least one (possibly infinitely many) vertex of level $i+1$ for all $i \ge 1$. \item If $|v| \ge 2$, the set of vertices of level $|v|-1$ joined with $v$ is finite and nonempty. \end{compactenum} We will also add a level 0 which consists of one point that has 1 edge connecting it to each point in $\Gamma_1$. Graded graphs are also called {\bf Bratelli diagrams} in the case that we assume that all levels are finite. \end{definition} \begin{example} For a chain of compact groups $G(1) \subset G(2) \subset G(3) \subset \cdots$, one can construct a graded graph as follows. The vertices of level $N$ are equivalence classes of complex irreducible representations of $G(N)$. Choose a representation $\pi_v$ in an equivalence class $v$. Then $v$ is joined with $u$ (with $|u|=|v|-1$) if $\pi_u$ is a subrepresentation of $\pi_v|_{G(N-1)}$. The multiplicity of the $uv$ edge is the multiplicity of $\pi_u$ in $\pi_v|_{G(N-1)}$. Note that $\dim \pi_v$ is finite for any irreducible representation, so the finiteness conditions for this graph are automatic. We call this the {\bf branching graph} of the above chain. Later we will consider the example $U(1) \subset U(2) \subset U(3) \subset \cdots$ where $U(N)$ is the unitary group of $N \times N$ matrices, and the inclusion $U(N) \subset U(N+1)$ is given by $A \mapsto \begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}$. This graph is called the {\bf Gelfand--Tsetlin graph}. Note that this is equivalent to considering representations of $\GL_1(\bC) \subset \GL_2(\bC) \subset \cdots$ via Weyl's unitary trick. \end{example} We define the {\bf dimension} of $v \in \Gamma$ to be the number of paths from level 1 to $v$ (counting multiplicities) which increase once by level at each step. For groups, assuming that $G(1)$ only has 1-dimensional representations, this definition of dimension coincides with the usual definition of dimension of a representation. Then we have the recurrence relations \[ \dim v = \sum_{u : |u| = |v| - 1} (\dim u)(\dim(u,v)) \] where $\dim(u,v)$ is the multiplicity of the edge from $u$ to $v$. We can define a projective chain by $V_N = \Gamma_N$ and $\Lambda_N^{N+1}(v,u) = \frac{(\dim u) (\dim(u,v))}{\dim v}$ for $v \in V_{N+1}$ and $u \in V_N$. More generally, $\Lambda^{N'}_N(v,u) = \frac{(\dim u) (\dim(u,v))}{\dim v}$ where $\dim(u,v)$ is the number of paths from $u$ to $v$. \begin{proposition} If $\Gamma$ is a branching graph of a chain coming from compact groups, then there is a natural bijection between the boundary and extreme continuous characters of $G(\infty) = \bigcup_{n \ge 1} G(n)$. \end{proposition} We denote paths as $v_1 e_{12} v_2 e_{23} v_3 \cdots $ where $v_i \in \Gamma_i$. Let $T$ denote the space of all infinite paths. \begin{definition} Let $P$ be a probability measure on $T$. Then $P$ is a {\bf Gibbs measure} if for any two finite initial paths $p_1,p_2$ with common end, we have that the probability of the set of paths starting with $p_1$ (cylindrical set) is the same as the probability of the set of paths starting with $p_2$. \end{definition} We remark that historically, the notion of Gibbs measures was instead referred to as central measures (Vershik--Kerov). \begin{proposition} There is a natural bijection between Gibbs measures on $T$ and coherent systems of measures on the projective chain given by $\Gamma$. \end{proposition} \begin{proof} The proof is the same as the one that was given earlier for Young's graph. \end{proof} Hence Gibbs measures are in bijection with probability measures on $V_\infty = \partial \Gamma$ (where $\partial$ means boundary). \subsection{The Martin boundary.} We continue to work with projective chains. Also, assume that $V_N$ are finite (for simplicity). For a vertex $v \in V_N$, define \[ f_v(v') = \begin{cases} \Lambda^N_{N'}(v,v') & \text{if } v' \in V_M \text{ and } M < N\\ 1 & \text{if } v' = v\\ 0 & \text{otherwise}\end{cases}. \] Let $\cF = \bR^{\coprod_{i \ge 1} V_i}$ denote the space of all real-valued functions on $V = \coprod_{i \ge 1} V_i$ with the topology given by pointwise convergence. Then $v \mapsto f_v$ gives an inclusion $V \to \cF$. Denote by $\ol{V}$ the closure of $V$ in $\cF$. The {\bf Martin boundary} is defined as $\ol{V} \setminus V$. Any element in $\ol{V} \setminus V$ defines a coherent system. \begin{theorem} For a projective system with finite $V_N$, we have $V_\infty \subseteq \ol{V} \setminus V$. \end{theorem} % 10/6/11 \subsection{Kingman graphs} Let $T_n$ be the set of monotone paths from $\emptyset$ to some vertex in level $n$. There are natural projections $\pi_{n,n-1} \colon T_n \to T_{n-1}$ obtained by removing the last edge. We define an equivalence relation on $T_n$ via $p \sim_n p'$ if they have the same final vertex. Then \begin{compactenum} \item The collection of $T_n$ with projections $\pi_{n,n-1}$ and the equivalence relations $\sim_n$ allows us to reconstruct the initial branching graph. For example $\Gamma_n = T_n / \sim_n$. \item Conversely, assume that $T_n / \sim_n$ is finite for all $n$. Then one can construct a locally finite branching graph given the data $(T_n, \pi_{n,n-1}, \sim_n)_n$ if and only if \begin{compactenum} \item For any equivalence classes $u \subset T_{n-1}$ and $v \subset T_n$, and any $t \in u$, we have $\#\{(\pi_{n,n-1})^{-1} t \cap v \}$ depends only on $u$ and $v$, and not the choice of $t \in u$. \item $\pi_{n,n-1}$ is surjective for all $n$. \end{compactenum} \end{compactenum} A {\bf partition of a set} $A$ is a splitting of $A$ into blocks. Let $T_n$ be the set of all partitions of $[n]:=\{1, \dots, n\}$ (by convention, $T_0$ is a singleton). If $A \supset B$, then a partition of $A$ induces a partition on $B$. In particular, the inclusion $[n-1] \subset [n]$ induces a projection $\pi_{n,n-1} \colon T_n \to T_{n-1}$. We say that two partitions of $n$ are equivalent if they are the same after applying a permutation of $n$. This defines a branching graph $\bK$. In particular, the vertices can be identified with Young diagrams. The edges are also the same as in $\bY$, except now they have multiplicities. The multiplicity of the edge $\mu \ne \lambda$ in $\bK$ is the number of rows in $\lambda$ which have the same length as the row that contains $\lambda \setminus \mu$. The dimension is given by a multinomial coefficient \[ \dim_\bK \lambda = \frac{|\lambda|!}{\lambda_1! \lambda_2! \cdots }. \] In this setup, Gibbs measures are also known as {\bf Kingman partition structures}. A branching graph is {\bf multiplicative} if there exists a $\bZ_+$-graded algebra $\cA = \bigoplus_{n \ge 0} \cA_n$ with $\dim \cA_0 = 1$ with a distinguished homogeneous basis $a_\lambda \in \cA_n$, ($\lambda \in \Gamma_n$, so $\dim A_n = |\Gamma_n|$), and a distinguished element $1 \in \Gamma_1$, such that $a_\lambda \cdot a_1 = \sum_{\nu \in \Gamma_{n+1}} \mult(\lambda, \nu) a_\nu$. \begin{example} \begin{enumerate} \item Pascal graph: take $\cA = \bC[x,y]$ with basis $\{x^k y^\ell\}$ and $a_1 = x+y$ \item Young graph: $\cA = \Lambda$ with basis $\{s_\lambda\}$, and $a_1 = s_1$ \item Kingman graph: $\cA = \Lambda$ with basis $\{m_\lambda\}$ and $a_1 = m_1$ \qedhere \end{enumerate} \end{example} \subsection{Gelfand--Tsetlin graph.} We now focus on the branching graph for $U(1) \subset U(2) \subset \cdots$. The irreducible complex representations of $U(n)$ are parametrized by {\bf signatures} (commonly known as {\bf highest weights}). A signature is a weakly decreasing sequence $\{\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_N \in \bZ^N\} = \bG\bT_N$. The branching rule is given by \[ \pi_\lambda |_{U(n-1)} = \bigoplus_{\substack{\mu \in \bG\bT_{n-1} \\ \mu \prec \lambda}} \pi_\mu \] where we define $\mu \prec \lambda$ if $\lambda_1 \ge \mu_1 \ge \lambda_2 \ge \mu_2 \ge \cdots \ge \mu_{N-1} \ge \lambda_N$. Characters of the $\pi_\lambda$ are rational Schur functions, since we have \[ \Tr(\pi_\lambda (\diag(u_1, \dots, u_N))) = \frac{\det( u_i^{N+\lambda_j - j})}{\det(u_i^{N-j})}. \] by Weyl's character formula. Then the branching rule is just stating the identity \begin{align} \label{eqn:GLbranching} s_\lambda(x_1, \dots, x_N,1) = \sum_{\mu \prec \lambda} s_\mu(x_1, \dots, x_{N-1}). \end{align} We can interpret this in terms of paths $\lambda^{(1)} \prec \lambda^{(2)} \prec \cdots$. This gives the interpretation of $s_\lambda$ as a generating function for semistandard Young tableaux. We can also get Weyl's dimension formula \[ s_\lambda(1, \dots, 1) = \prod_{i < j} \frac{\lambda_i - i - (\lambda_j - j)}{j-i}. \] Another interpretation: consider a path $(1) \prec (2,0) \prec (3,2,-1)$. Replace $\lambda^{(j)}_i$ by $\lambda^{(j)}_i + \frac{j}{2} - i$. Then the above becomes $(\frac{1}{2}) \prec (2, -1) \prec (\frac{7}{2}, \frac{3}{2}, -\frac{5}{2})$. For each coordinate $x$ in $\lambda^{(j)}$, plot the point $(x,j)$ in the plane. Surround each point by a diamond whose vertices are $(x \pm \frac{1}{2}, j \pm 1)$. This can be completed to a tiling of the plane using parallelograms. This gives a 3-d picture which one can imagine as a plane partition which gives rise to a reverse semistandard Young tableau. The boundary $\Omega_{\bG\bT}$ of $\bG\bT$ is parametrized by points $(\alpha^+, \beta^+, \delta^+, \alpha^-, \beta^-, \delta^-) \in \bR_+^\infty \times \bR_+^\infty \times \bR_+ \times \bR_+^\infty \times \bR_+^\infty \times \bR_+$ subject to the conditions \begin{align*} \alpha_1^+ \ge \alpha_2^+ \ge \cdots \ge 0,\\ \beta_1^+ \ge \beta_2^+ \ge \cdots \ge 0,\\ \alpha_1^- \ge \alpha_2^- \ge \cdots \ge 0,\\ \beta_1^- \ge \beta_2^- \ge \cdots \ge 0,\\ \beta_1^+ + \beta_1^- \le 1,\\ \sum_{j=1}^\infty(\alpha_j^\pm + \beta_j^\pm) \le \delta^\pm. \end{align*} We let $\gamma^\pm = \delta^\pm - \sum_{j=1}^\infty (\alpha_j^\pm + \beta_j^\pm)$. The set $\Omega_{\bG\bT}$ is locally compact and for any $C \in \bR$, the subsets defined by $\{ \delta^+ + \delta^- \le C\}$ are compact. In order to prove this, we need to construct the links $\Lambda^\infty_N \colon \Omega_{\bG\bT} \to \bG\bT_N$. Given $u \in \bC$ with $|u|=1$ and $\omega \in \Omega_{\bG\bT}$, define \begin{align} \label{eqn:GTPhi} \Phi(u,\omega) = \exp(\gamma^+(u-1) + \gamma^-(u^{-1} - 1)) \prod_{i \ge 1} \frac{1 + \beta_i^+(u-1)}{1-\alpha^+_i(u-1)} \frac{1+\beta^-_i(u^{-1}-1)}{1 - \alpha_i^-(u^{-1}-1)}. \end{align} For each $\omega \in \Omega_{\bG\bT}$, this is a function which is holomorphic in a neighborhood of the unit circle. Write its Laurent series expansion \[ \Omega(u,\omega) = \sum_{n \in \bZ} \phi_n(\omega) u^n. \] Then the $\phi_n$ give functions on $\Omega_{\bG\bT}$. \begin{lemma} \begin{compactenum}[{\rm 1.}] \item $|\Phi(u,\omega)| \le 1$ for all $|u|=1$ \item $\Phi(u,\omega)$ determines $\omega \in \Omega_{\bG\bT}$ uniquely. \end{compactenum} \end{lemma} Now define \[ \Lambda^\infty_N(\omega, \lambda) = (\dim_{\bG\bT} \lambda) \det(\phi_{\lambda_i - i + j}(\omega))_{i,j=1}^N. \] The determinant is a harmonic function on $\bG\bT$. It is useful to compare this determinant to the definition of $s_\lambda^\circ(\omega) = \det(h^\circ_{\lambda_i - i + j}(\omega))$ used for Young's graph. In the latter case, it is a minor of an upper unitriangular Toeplitz matrix, and in the former case, it is a minor of a general Toeplitz matrix. Recall also the generating function \[ \sum_{n \ge 0} h^\circ_n u^n = \exp(\gamma u) \prod_{i \ge 1} \frac{1+ \beta_i u}{1 - \alpha_i u}. \] \begin{proposition} $\Lambda^\infty_N$ is a Markov kernel and we have $\Lambda_N^\infty \Lambda_M^N = \Lambda_M^\infty$ for all $N>M$. \end{proposition} Recall that we had embeddings $\bY_n \subset \partial \bY$. The way to do this was to use modified Frobenius coordinates. We can define an embedding $\bG\bT_N \subset \partial \bG \bT$ as follows. Given a signature, we get two Young diagrams $(\lambda^+, \lambda^-)$ (the first given by the nonnegative coordinates and the second given by the negative coordinates, but it is drawn upside down). Now write $\lambda^\pm = (a_1^\pm, a_2^\pm, \dots | b_1^\pm, b_2^\pm, \dots)$ in modified Frobenius coordinates. Now set $\alpha_i^\pm = a^\pm_i / N$, $\beta_i^\pm = b_i^\pm / N$, and $\delta^\pm = |\lambda^\pm| / N$. %10/13/2011 This also answers the question of finding all possible limits of $\Tr(\pi^\lambda(-)) / \dim_N \lambda$ where $\pi^\lambda$ is an irreducible representation of $U(N)$, as $N \to \infty$. These are functions of the form \eqref{eqn:GTPhi}. The full list of extreme characters of $U(\infty)$ was given by D. Voiculescu in 1974 and also Edrei in 1953. The approximation approach was carried out by Okounkov--Olshanski in 1998, i.e., computing the limits $(\Tr(\pi^\lambda(-)))/ \dim \lambda$. In 2011, a proof by Borodin--Olshanski was given for the asymptotics of $\dim(\mu,\lambda) / \dim(\lambda)$. \begin{remark} Characters (as we have defined them) of $S_\infty$ or $U(\infty)$ do correspond to actual representations. Given an extreme character $\chi \colon G \to \bC$, where $G$ is $S_\infty$ or $U(\infty)$, corresponds to a homomorphism $T \colon G \to U(\cH)$ where $\cH$ is a Hilbert space. The closure of $T(G)$ in the weak operator topology gives an {\bf operator (von Neumann) algebra} $\cA$. Let $\cA^!$ be the space of operators that commute with $\cA$. Then being an extreme character corresponds to the statement $\cA \cap \cA^! = \bC$. The operator algebras $\cA$ satisfying this condition are called {\bf factors}. Factors have types I, II, or III. Those of type I are of the form $B(\cH)$ (bounded operators on a Hilbert space $\cH$), and those of type II admit a trace function. Either this trace takes values in $[0, \infty)$ or $[0,1)$, and the second kind correspond to extreme characters. \begin{example} $S_\infty$ acts on $\ell^2(S_\infty)$ via $(T(\sigma)f)(\tau) = f(\sigma^{-1}\tau)$. This is a unitary representation of $S_\infty$. The operator algebra $\cA = \ol{T(S_\infty)}$ is a factor of type ${\rm II}_1$. Equivalently, $S_\infty \times S_\infty$ acts on $\ell^2(S_\infty)$ via $(T(\sigma_1, \sigma_2) f)(\tau) = f(\sigma_1^{-1} \tau \sigma)$ and this representation is irreducible. In this case, we would have $\cA^! = \ol{T(-,e)}$. This representation corresponds to the case $\{\alpha_i = \beta_j = 0\}$. \end{example} Possible reference: G. Olshanski's work \end{remark} The structure of the Gelfand--Tsetlin graph mimics the branching rule. We define the link \[ \Lambda_N^{N+1}(\lambda, \mu) = \frac{(\dim_N \mu) (\dim_{N,N+1}(\mu,\lambda))}{\dim_{N+1}\lambda}. \] Recall that branching formula \eqref{eqn:GLbranching}. More generally, we have \[ s_\lambda(x_1, \dots, x_N, a) = \sum_{\mu \prec \lambda} s_\mu(x_1, \dots, x_N) a^{|\lambda| - |\mu|}. \] Then we can modify our dimension function by defining $\tilde{\dim}_N(\nu) = s_\nu(a_1, \dots, a_N)$ and similarly define $\tilde{\Lambda}^{N+1}_N$. It is widely unknown what the boundary looks like. However, in the $a_j = q^j$, this was worked out by Gorin in 2005. In this case, the boundary is $\{ \eps_1 \le \eps_2 \le \cdots \mid \eps_i \in \bZ\}$. One could also try to replace Schur functions by Hall--Littlewood polynomials or Macdonald polynomials, but the boundary is also mostly unknown here. \subsubsection{Semiclassical limit} We can ``scale'' the Gelfand--Tsetlin graph and consider the limit. This limit describes the following ``boundary'': the representations of $U(N)$ are replaced by $U(N)$-invariant probability measures on $H_N = \{H \in {\rm Mat}(N,\bC) \mid H^* = H \}$, where the action of $U(N)$ on $H_N$ is by conjugation. Recall that Hermitian matrices are diagonalizable, so two such are equivalent under conjugation if and only if they have the same eigenvalues. So the extreme points are {\bf orbital measures}: given $(x_1 \ge \dots \ge x_N) \in \bR^N$, we define the orbital measure $\cO_{(x_1 \ge \dots \ge x_N)}$ to be the pushforward of the Haar measure on $U(N)$ via $U \mapsto U^{-1} {\rm diag}(x_1, \dots, x_N) U$. The boundary is the set of ergodic / extreme $U(\infty)$-invariant measures on $H_\infty = \{(H_{ij})_{i,j \ge 1} \mid H^* = H\}$. The characters become integrals $\int_{H_\infty} \exp(i \Tr(Y H) \mu({\rm d} H)$ where $Y = {\rm diag}(y_1, y_2, \dots)$. For extreme measures, this integral is of the form $\prod_{j \ge 1} F(y_j)$ where \[ F(y) = \exp(-\gamma_1 y^2 + \gamma_2 y) \prod_{i \ge 1} \frac{ \exp^{-i y \alpha_k}}{1 - i y \alpha_k} \] for some $\gamma_1 \ge 0$, $\gamma_2 \in \bR$, $\alpha_k \in \bR$, and $\sum_k \alpha_k^{-2} < \infty$. This can be interpreted as a limit of the functions $\Phi$ in \eqref{eqn:GTPhi}. \subsubsection{Young bouquet.} What about the connection between $U(\infty)$ and $S_\infty$? Or rather, the connection between $\bG\bT$ and $\bY$? There is a connecting object $\bG\bT \to \bY\bB$ where $\bY\bB$ is the ``Young bouquet''. To define this, we first define the binomial projective system $\bB$. Let $I = \bR_{> 0}$ and $V_r = \bZ_{\ge 0}$ for all $r \in I$. The links are defined by \[ \leftexp{\bB}{\Lambda_r^{r'}}(n,m) = \left(1-\frac{r}{r'}\right)^{n-m} \left(\frac{r}{r'}\right)^m \frac{n!}{(n-m)! m!} \] where by convention, dividing by a negative factorial results in 0, i.e., $1/k! = 0$ for $k<0$. If we have $n$ i.i.d. uniform points $z_1, \dots, z_n$ on $[0,r']$, then $\Lambda^{r'}_r(n,m) = {\rm Prob}\{\#\{z_j \in [0,r]\} = m\}$. The boundary $\partial \bB$ is $\bR_{\ge 0}$ with the Poisson distribution \[ \leftexp{\bB}{\Lambda_r^\infty}(x,m) = \exp(-xr) \frac{(xr)^m}{m!}. \] So boundary points are uniform Poisson processes on $I = \bR_{\ge 0}$ of density $x \in \bR_{\ge 0}$. Now define $\bY\bB_r = \bY$ for $r > 0$ and $\bY\bB_0 = \{\emptyset\}$. The links are defined by \[ \leftexp{\bY\bB}{\Lambda_r^{r'}}(\lambda, \mu) = \leftexp{\bB}{\Lambda_r^{r'}}(|\lambda|, |\mu|) \leftexp{\bY}{\lambda_{|\mu|}^{|\lambda|}}(\lambda, \mu). \] Paths in $\bY$ are standard Young tableau and paths in $\bB$ are locally constant functions $\bR_{\ge 0} \to \bR_{\ge 0}$ with jumps of size 1. Paths in $\bY\bB$ are ``overlaps'': locally constant functions $f \colon \bR_{\ge 0} \to \bY$ with $f(0) = \emptyset$ and jumps by 1 at the ``discontinuities''. One can interpret $\bY\bB$ as a limit of $\bG\bT$. First note that paths in $\bG\bT$ are semistandard Young tableaux. Let $\mu$ be a fixed Young diagram. We can think of $\mu \in \bG\bT_N$ with $N \ge \ell(\mu)$. Then paths from $\emptyset$ to $\mu \in \bG\bT_N$ are SSYT using the numbers $1, \dots, N$. The Gibbs measure property says that one has to consider all such SSYT with a uniform measure. For large $N$, all of the labels will typically be distinct. If we normalize by dividing by $N$, we get something like a path in $\bY\bB$ (except the entries are restricted to be in $[0,1]$). A more precise statement: fix $\nu$ and $\mu$, then \[ \lim_{N,N' \to \infty} \leftexp{\bG\bT}{\Lambda_N^{N'}}(\nu, \mu) = \leftexp{\bY\bB}{\Lambda_r^{r'}}(\nu, \mu) \] where $N'/N \to r'/r$. So one can think of $\bY\bB$ as an extension of $\bY$ and as a limit of $\bG\bT$. % The boundary $\partial \bY\bB$ will be an extension of $\partial \bY$ % and a limit of $\partial \bG \bT$. %10/18/2011 The boundary is given by \[ \partial \bY\bB = \{ (\tilde{\alpha}, \tilde{\beta}, \tilde{\delta}) \mid \tilde{\alpha}_1 \ge \tilde{\alpha}_2 \ge \cdots \ge 0,\ \tilde{\beta}_1 \ge \tilde{\beta}_2 \ge \cdots \ge 0,\ \sum_i (\tilde{\alpha}_i + \tilde{\beta}_i) < \tilde{\delta} \}. \] We can rewrite $\partial \bY\bB$ as $\{(0,0,0)\}$ union the cone over $\partial \bY$. Namely, to $(\tilde{\alpha}, \tilde{\beta}, \tilde{\gamma}) \in \partial \bY\bB$, we associate $(\tilde{\alpha} / \tilde{\delta}, \tilde{\beta} / \tilde{\delta}, \delta) \in \partial \bY \times \bR_{>0}$. For $(\alpha, \beta) \in \partial \bY$, we have \[ \Lambda^\infty_N((\alpha, \beta); \lambda) = (\dim \lambda) s_\lambda^\circ(\alpha, \beta). \] Also we have \begin{align*} \Lambda_N^\infty(\lambda) &= (\dim_N \lambda) \det( \phi_{\lambda_i - i + j})_{i,j=1}^N\\ \sum_m \phi_m u^m &= \exp(\gamma^+(u-1) + \gamma^-(u^{-1} - 1)) \prod_i \frac{(1 + \beta_i^+(u-1))(1+\beta^-_i(u^{-1}-1))}{(1 - \alpha_i^+(u-1))( 1 - \alpha_i^-(u^{-1}-1))}. \end{align*} And for $((\alpha, \beta), \delta) \in \partial \bY \times \bR_{>0}$, we have \[ \Lambda^\infty_r (((\alpha, \beta), \delta), \lambda) = \exp(-r\delta) \frac{(r\delta)^{|\lambda|}}{|\lambda|!} (\dim \lambda) s_\lambda^\circ(\alpha, \beta). \] There is a direct transition \[ \Lambda_{\lfloor r / \eps \rfloor}^\infty (\alpha^+ = \eps \tilde{\alpha}, \beta^+ = \eps \tilde{\beta}, \delta^+ = \eps \tilde{\delta}, \alpha^- = 0, \beta^- = 0, \gamma^- = 0; \lambda) \xrightarrow{\eps \to 0} \Lambda_r^\infty((\tilde{\alpha}, \tilde{\beta}, \tilde{\delta}, \lambda)). \] \section{Random point processes.} What if we want a decomposition on extreme characters (i.e., a measure on $\partial \bY$ or $\partial \bG\bT$, etc.)? How do we define measures on infinite-dimensional spaces? Let $\cX$ be a locally compact and separable space, such as $\bZ^d$, $\bR^d$. Then $\cX$ is a {\bf state space}, and we refer to elements of $\cX$ as {\bf particles}. A {\bf point configuration} in $\cX$ is a locally finite subset (i.e., no accumulation points in $\cX$). Define $\Conf(\cX)$ to be the space of point configurations in $\cX$. A relatively compact (i.e., closure is compact) subset $A$ of $\cX$ is called a {\bf window}. For a window $A$, define $N_A \colon \Conf(\cX) \to \bZ_{\ge 0}$ by $N_A(X) = \#(X \cap A)$. We put a Borel structure on $\Conf(\cX)$ which is generated by these functions, i.e., the smallest Borel structure which makes these functions measurable. A {\bf random point process} is a probability measure on $\Conf(\cX)$. \subsection{Examples.} \subsubsection{Poisson process.} Take $\cX = \bR$. The (uniform) {\bf Poisson process} is characterized by 2 properties: \begin{compactitem} \item $N_A$ is Poisson distributed for any window $A$ as $P(N_A = k) = e^{-|A|} |A|^k / k!$. \item For any two disjoint windows $A$ and $B$, $N_A$ and $N_B$ are independent (no interaction). \end{compactitem} How to construct the Poisson process? Take $M \gg 0$, $M \in \bZ$. Consider $M$ i.i.d. uniform random variables on $[-M/2, M/2]$. Then there is approximately 1 point per unit length. For $M \to \infty$, the limit exists and it is the uniform Poisson process. The $M \to \infty$ picture is simpler than the case of $M$ finite. For instance in the finite case, the probability for the number of points in a window is not as clean as an exponential function, and will be messy, but it limits to the exponential function. The standard definition is that the distances between neighboring particles are i.i.d. according to $e^{-x} {\rm d}x$. To define a non-uniform Poisson process with respect to a measure $\mu$, we change the probabilities by $P(N_A = k) = e^{-\mu(A)} \mu(A)^k / k!$. This definition makes sense for any measure space. \subsubsection{Bernoulli process.} Take $\cX = \bZ$ and $p \in [0,1]$. The probability of containing a given particle is $p$, and the probability of not containing it is $1-p$. Different locations are clearly independent. Now embed $\bZ \to \bR$ via $m \mapsto pm$. With this embedding, as $p \to 0$, the Bernoulli process converges to uniform Poisson. Convergence means the following: for any set of windows $A_1, \dots, A_m$, we have that $(N_{A_1}, \dots, N_{A_m})$ converges in distribution. \subsection{Correlation functions.} Start with $\cX$ discrete with a point process. For a finite subset $A \subset \cX$, define the {\bf correlation function} of $A$ as \[ \rho(A) = \Prob\{X \in \Conf(\cX) \mid X \supset A\}. \] If $A = \{x_1, \dots, x_n\}$, we also write $\rho(A) = \rho_n(x_1, \dots, x_n)$. This is a symmetric function / measure on $\cX^n$. \begin{example} For the Bernoulli process, $\rho_n(x_1, \dots, x_n) = p^n$. \end{example} \begin{proposition} Let $\cX$ be discrete. The sequence of correlation functions $(\rho_1, \rho_2, \dots)$ determines the point process on $\cX$ uniquely. \end{proposition} \begin{proof} A point process is a probability measure on $\{0,1\}^\cX$. A base for the topology / Borel structure is formed by sets $U_A^B = \{ X \in \Conf(\cX) \mid X \cap B = A\}$ where $A \subset B$ are finite. A probability measure $P$ on $\{0,1\}^\cX$ is uniquely determined by $\{P(U_A^B)\}_{A \subset B \text{ finite}}$. But explicitly, we have \[ P(U_A^B) = \sum_{C : A \subset C \subset B} (-1)^{|C|-|A|} \rho(C) \] by inclusion-exclusion. \end{proof} How do we extend this definition of correlation function for continuous $\cX$? For any $n = 1, 2, \dots$, define the {\bf $n$th correlation measure} $\rho_n$ on $\cX^n$, which is characterized by the following property: for any test function $F \colon \cX^n \to \bR$, one has \[ \int_{\cX^n} F(x_1, \dots, x_n) \rho_n({\rm d}x_1 \cdots {\rm d}x_n) = \bE_{\substack{X = \{x_1, x_2, \dots\} \\ \in \Conf(\cX)}} \sum_{\substack{i_1, \dots, i_n \\ \text{pairwise distinct}}} F(x_{i_1}, \dots, x_{i_n}). \] Here $\bE$ denotes expected value. For $\cX$ discrete, these two definitions agree. Typically, a state space $\cX$ comes with a reference measure $\mu$ (counting measure for discrete $\cX$, Lebesgue measure on $\bR$, etc.). The density of $\rho_n({\rm d}x_1 \dots {\rm d}x_n)$ with respect to $\mu^{\otimes n}$ is called the {\bf $n$th correlation function}, where \[ \rho_n({\rm d}x_1 \cdots {\rm d}x_n) = \rho_n(x_1, \dots, x_n) \mu({\rm d}x_1) \cdots \mu({\rm d}x_n). \] For the Poisson process, $\rho_n(x_1, \dots, x_n) = 1$. For $\cX \subset \bR$, \[ \rho_n(x_1, \dots, x_n) = \lim_{\Delta x_1, \dots, \Delta x_n \to 0} \frac{\Prob \{ X \text{ has a particle in each of } [x_i, x_i + \Delta]\}}{ \Delta x_1 \cdots \Delta x_n}. \] Consider point processes with $N$ points given by a probability measure (joint probability distribution) $P({\rm d}x_1 \dots {\rm d}x_N)$ which is symmetric with respect to permutations. Then \[ \rho_n({\rm d}x_1 \dots {\rm d}x_n) = \frac{N!}{(N-n)!} \int_{x_{n+1}, \dots, x_n \in \cX} P({\rm d}x_1 \dots {\rm d}x_N). \] To see this, use the definition and note that \begin{align*} {\bf E} \sum_{\substack{i_1, \dots, i_n \\ \text{pairwise distinct}}} F(x_{i_1}, \dots, x_{i_n}) &= \int \sum_{\substack{i_1, \dots, i_n\\ \text{pairwise distinct}}} F(x_{i_1}, \dots, x_{i_n}) P({\rm d}x_1 \dots {\rm d}x_n)\\ &= \frac{N!}{(N-n)!} \int F(x_1, \dots, x_n) P({\rm d}x_1 \dots {\rm d}x_n)\\ &= \int_{\cX^n} F(x) \rho_n({\rm d}x). \end{align*} \subsection{Determinantal point processes.} \begin{definition} A point process on $\cX$ is {\bf determinantal} if there is a kernel $K \colon \cX \times \cX \to \bC$ such that \[ \rho_n(x_1, \dots, x_n) = \det(K(x_i, x_j))_{i,j=1}^n. \] $K(x,y)$ is called the {\bf correlation kernel}. \end{definition} Some small examples: $\rho_1(x) = K(x,x)$ and $\rho_2(x,y) = \rho_1(x) \rho_1(y) - K(x,y)K(y,x)$. \begin{definition} Consider a state space $\cX$ with a reference measure $\mu$. An $N$-point {\bf biorthogonal ensemble} on $\cX$ is an $N$-point point process (probability measure on $\cX^N$) of the form \[ P_N({\rm d}x_1 \dots {\rm d}x_n) = c_N \det(\phi_i(x_j))_{i,j=1}^N \det(\psi_i(x_j))_{i,j=1}^N \mu({\rm d}x_1) \cdots \mu({\rm d}x_n) \] where $c_N$ is some constant and $\phi_i, \psi_i$ are arbitrary functions on $\cX$. \end{definition} \begin{proposition} Any biorthogonal ensemble is a determinantal point process. Its correlation kernel has the form \[ K(x,y) = \sum_{i,j=1}^N (G^{-T})_{i,j} \phi_i(x) \psi_j(y) \] where $G_{i,j} = \int_{\cX} \phi_i(x) \psi_j(x) \mu({\rm d}x)$ is the Gram matrix. \end{proposition} %10/20/2011 This was first considered by F. Dyson in 1962. \begin{proof} \begin{align*} \int_{\cX^N} \det(\phi_i(x_j))_{i,j=1}^N \det(\psi_i(x_j))_{i,j=1}^N {\rm d}x &= \int_{\cX^N} \sum_{\sigma, \tau \in S_N} \sgn(\sigma \tau) \prod_{i=1}^N \phi_{\sigma(i)}(x_i) \psi_{\tau(i)}(x_i) {\rm d}x\\ &= \sum_{\sigma, \tau \in S_N} \prod_{i=1}^N G_{\sigma(i), \tau(i)} \\ &= N! \sum_{\rho \in S_N} \sgn(\sigma) \prod_{i=1}^N G_{i, \rho(i)} = N! \det(G_{i,j})_{i,j=1}^N. \end{align*} This implies that $G$ is invertible, and we will have $c_N = (N! \det(G))^{-1}$. Now we have \begin{align*} \rho_n(x_1, \dots, x_n) = \frac{N!}{(N-n)!} \int_{(x_{n+1}, \dots, x_{N}) \in \cX^{N-n}} \det(\phi_i(x_j))_{i,j=1}^N \det(\psi_i(x_j))_{i,j=1}^N {\rm d}x \frac{1}{N! \det(G)}. \end{align*} Take matrices $A$ and $B$ such that $AGB^T = 1$. Set \[ \Phi_k = \sum_{\ell = 1}^N A_{k\ell} \phi_\ell, \quad \Psi_k = \sum_{\ell=1}^N B_{k\ell} \psi_\ell. \] Then we have \[ \langle \Phi_i, \Psi_j \rangle := \int_{\cX} \Phi_i(x) \Psi_j(x) {\rm d}x = [AGB^T]_{i,j} = 1_N. \] Also we have \begin{align*} \det(\Phi_i(x_j))_{i,j=1}^N &= \det(A) \det(\phi_i(x_j))_{i,j=1}^N, \\ \det(\Psi_i(x_j))_{i,j=1}^N &= \det(B) \det(\psi_i(x_j))_{i,j=1}^N. \end{align*} The result is that \[ P_N({\rm d}x_1 \cdots {\rm d}x_N) = \frac{1}{N!} \det(\Phi_i(x_j))_{i,j=1}^N \det(\Psi_i(x_j))_{i,j=1}^N, \] and \begin{align*} \rho_n(x_1, \dots, x_n) &= \frac{1}{(N-n)!} \int_{x_1, \dots, x_n} \det(\phi_i(x_j))_{i,j=1}^N \det(\psi_i(x_j))_{i,j=1}^N {\rm d}x\\ &= \frac{1}{(N-n)!} \int_{x_{n+1}, \dots, x_N} \sum_{\sigma, \tau \in S_N} \sgn(\sigma\tau) \prod_{i=1}^N \Phi_{\sigma(i)}(x_i) \Psi_{\tau(i)}(x_i) {\rm d}x_i\\ &= \frac{1}{(N-n)!} \int_{x_{n+1}, \dots, x_N} \sum_{\substack{\sigma, \tau \in S_N \\ \sigma(k) = \tau(k)\\ \text{for } k=n+1,\dots,N}} \sgn(\sigma\tau) \prod_{i=1}^n \Phi_{\sigma(i)}(x_i) \Psi_{\tau(i)}(x_i) {\rm d}x_i\\ &= \sum_{1 \le j_1 < \cdots < j_n \le N} \det \Phi^{j_1, \dots, j_n} \det \Psi^{j_1, \dots, j_n}, \end{align*} where $\Phi^{j_1, \dots, j_n}$ is the submatrix of $[\phi_i(x_j)]$ with columns $j_1, \dots, j_n$ and rows $i = 1, \dots, n$. Now using the Cauchy--Binet theorem, this last expression becomes \[ \det(\Phi \Psi^T)_{i,j=1}^n = \det\left(\sum_{k=1}^N \Phi_k(x_i) \Psi_k(j)\right)_{i,j=1}^N. \] Now define $K(x_i, x_j) = \sum_{k=1}^N \Phi_k(x_i) \Psi_k(x_j)$. We can write this as \begin{align*} \sum_{k=1}^N \Phi_k(x) \Psi_k(y) &= \sum_{k,\ell,m} A_{k\ell} \phi_\ell(x) B_{km}\psi_m(y)\\ &= \sum_{\ell, m} \phi_\ell(x) \psi_m(y) \sum_k A_{k\ell} B_{km} = A^TB = G^{-T}. \qedhere \end{align*} \end{proof} \begin{remark} A correlation kernel of a determinantal process is not defined uniquely. Namely, $K(x,y) \to \frac{f(x)}{f(y)} K(x,y)$ does not change $\det(K(x_i,x_j))$. \end{remark} \begin{remark} \begin{compactenum}[(a)] \item Taking limits does not destroy determinantal structure. \item There is a particle-hole involution on a discrete $\cX$ that preserves the class of determinantal processes. Namely, fix $Y$ be a subset of $\cX$. Given a point process, one can complement it within $Y$ (i.e., swap the notions of being or not being in $Y$) to get another point process. More formally, if the original process has kernel $K$, then the modified process has kernel $\begin{pmatrix} 1 - K & -K \\ K & K \end{pmatrix}$ where the block indices correspond to $Y$ and $\cX - Y$. \item Killing particles independently with some rate depending on location does not destroy the determinantal structure. \item Consider discrete $\cX$ and a Markov chain in $\cX$ with no loops. This induces a measure on point configurations given by trajectories, and such a point process is determinantal. \item Consider $\cX = [a,b] \subset \bZ$. A point process which is 1-dependent, i.e., its intersection with windows that have distance $\ge 2$ are independent, is determinantal. \qedhere \end{compactenum} \end{remark} \subsection{Orthogonal polynomial ensembles.} Take $\cX = \bR$. Let $w({\rm d}x)$ be a positive measure on $\cR$ with finite moments, i.e., $\int_{\bR} |x|^k w({\rm d}x) < \infty$ for all $k \ge 0$. \begin{example} The natural map $\bC[x]_{\le N} \to L^2(\bR, w({\rm d}x))$ is an embedding if and only if $\#\supp(w) > N+1$. \end{example} We will assume that $\#\supp(w) = \infty$. Notation: $V_N(x_1, \dots, x_n) = \prod_{1 \le i < j \le N} (x_i - x_j) = \det(x_i^{N-j})_{i,j=1}^N$. \begin{definition} The {\bf $N$-particle orthogonal polynomial ensemble} with weight $w$ is the $N$-point random point process with joint probability density \[ P_N({\rm d}x_1 \cdots {\rm d}x_N) = (V_N(x_1, \dots, x_N))^2 \prod_{i=1}^N w({\rm d}x_i). \qedhere \] \end{definition} \begin{example} The most well-known example is the {\bf Gaussian unitary ensemble} (GUE($N$)). In this case the space is $\{H \in {\rm Mat}_N(\bC) \mid H = H^*\}$ with measure $c e^{-\Tr(H^2)}$ for some constant $c$ \steven{fill in later}. The eigenvalues of $H$ form an $N$-point orthogonal polynomial ensemble with $w({\rm d}x) = e^{-x^2}{\rm d}x$. \end{example} An orthogonal polynomial ensemble is a biorthogonal ensemble with $\phi_i(x) = \psi_i(x) = x^{i-1} \sqrt{w(x)}$ where $w$ denotes the density function of the measure $w$. A kernel $K$ defines a linear operator $K \colon L^2 \to L^2$ by $(Kf)(x) = \int K(x,y) f(y){\rm d}y$. \begin{proposition} The correlation kernel $K(x,y)$ is the kernel of the orthogonal projection operator onto ${\rm span}(\sqrt{w(x)}, x\sqrt{w(x)}, \dots, x^{N-1}\sqrt{w(x)})$ in $L^2(\bR, w)$. \end{proposition} \begin{proof} Let $U = {\rm span}(\phi_j)_{j=1}^N$ and $V = {\rm span}(\psi_j)_{j=1}^N$ in $L^2(\bR, w)$. We have \[ \int \sum_{i,j=1}^N (G^{-T})_{i,j} \phi_i(x) \psi_j(y) \phi_m(y) {\rm d} y = \sum_{i,j=1}^N (G^{-T})_{i,j} G_{mj}^T \phi_i(x) = \sum_{i=1}^N 1_{i,m} \phi_i(x) = \phi_m(x). \] Hence $K|_U = 1_U$. On the other hand $K|_{V^\perp} = 0$. Also, one can check that $K^2 = K$. So $K$ is a projection onto $U$ which is parallel to $V^\perp$. In the orthogonal polynomial ensemble, we have $U = V$. \end{proof} \begin{definition} A {\bf system of orthogonal polynomials} on $\bR$ with weight $w$ is a sequence $\{p_n(x)\}_{n \ge 0}$ with $p_n \in \bC[x]$ and $\deg p_n = n$, such that $p_n \perp \bC[x]_{\le n-1}$ in $L^2(\bR,w)$, i.e., \[ \int_{\bR} p_n(x) p_m(x) w({\rm d}x) = \| p_n\|^2 \delta_{m,n}. \qedhere \] \end{definition} Note that a system of orthogonal polynomials is an orthogonal basis in $\bC[x]$ with inner product $\langle f,g \rangle = \int_{\bR} f(x) g(x) w({\rm d}x)$, so one can construct such systems using the Gram--Schmidt orthogonalization algorithm. Note that the degree constraint $\deg p_n = n$ uniquely determines the $p_n$ up to a constant. Standard notation: Let $k_n$ be the leading coefficient of $p_n$ and set $h_n = \| p_n\|^2_{L^2(\bR,w)}$. \begin{proposition} Let $\{p_n\}$ be the sequence of monic orthogonal polynomials with weight $w$. Then the correlation kernel of the $N$-point orthogonal polynomial ensemble has the form \[ K_N(x,y) = \sum_{j=0}^{N-1} \frac{p_j(x) p_j(y)}{h_j} \] with respect to the reference measure $w({\rm d}x)$ on $\bR$. \end{proposition} \begin{proof} Let $\pi_{j-1}$ be monic polynomials of degree $j-1$. Then \begin{align*} P_N({\rm d}x_1 \cdots {\rm d}x_N) &= c \det(x_i^{j-1}) \det(x_i^{j-1}) \prod w({\rm d}x_i)\\ &= \det(\pi_{j-1}(x_i)) \det(\pi_{j-1}(x_i)) \prod w({\rm d}x_i), \end{align*} where the last equality is via row operations. Then $G^{-T} = {\rm diag}(h_0^{-1}, \dots, h_{N-1}^{-1})$. \end{proof} \begin{proposition}[Christoffel--Darboux] \[ \sum_{j=0}^{N-1} \frac{p_j(x) p_j(y)}{h_j} = \frac{p_N(x) p_{N-1}(y) - p_{N-1}(x) p_N(y)}{h_{N-1}(x-y)}. \] \end{proposition} % 10/27/11 \begin{proof}[Proof 1.] Consider the operator given by multiplication by $x$ in $\bR[x]$. This operator is self-adjoint: \[ (x \cdot f, g)_{L^2(\bR,\omega)} = \int x f(x) g(x) w({\rm d}x) = (f, x \cdot g)_{L^2(\bR, \omega)}. \] The matrix of a self-adjoint operator in any orthonormal basis is symmetric. For our basis, we will use $\{p_n / \|p_n\|\}_{n \ge 0}$. By degree considerations, this matrix must be 0 below the subdiagonal. By symmetry, it must be 0 above the superdiagonal. Since the $p_n$ are monic, we have \[ x p_n = A_{n,n+1} p_{n+1} + A_{n,n} p_n + A_{n,n-1} p_{n-1} \] where $A_{n,n+1} = 1$. Now multiply the desired identity by $(x-y)$ and use this recurrence relation and symmetry of the matrix to finish (the left-hand side is a telescoping sum). \end{proof} \begin{proof}[Proof 2.] Consider the average ($x_1, \dots, x_N$ distributed as the orthogonal polynomial ensemble). Then \[ \bE ( \prod_{i=1}^N (u-x_i)(v-x_i)) = {\rm constant} \cdot \int \prod_{i=1}^N (u-x_i)(v-x_i) \prod_{i < j} (x_i-x_j)^2 w({\rm d}x_1) \cdots w({\rm d}x_N). \] (If the $x_i$ are eigenvalues of a random matrix $X$, then $\prod_i (u-x_i) = \det(u-X)$.) In the simpler case, we have \begin{align*} \bE(\prod_{i=1}^N (u-x_i)) &= {\rm constant} \cdot \int \prod_{i=1}^N (u-x_i) \prod_{i0$. Then we have $s_\lambda \mapsto \frac{\dim \lambda}{|\lambda|!} \theta^{|\lambda|}$. This is called the {\bf Plancherel specialization}. \begin{example} Recall the Young bouquet: the levels are marked by $r>0$ with boundary the set of all $\{(\alpha_i, \beta_i)\}$ satisfying some conditions. For any such $(\alpha, \beta)$, the measure on level $r$ had the form $\Prob(\lambda) = s_\lambda(\alpha, \beta) = s_\lambda(\alpha; \beta) \frac{\dim \lambda}{|\lambda|!} r^{|\lambda|} c$ where $c$ is some factor. \end{example} Taking $\pi_x$ and $\pi_y$ to be Plancherel specializations, we get \[ \Prob(\lambda) = % \frac{(\frac{\dim \lambda}{|\lambda|!} \theta^{|\lambda|})^2}{{\rm % constant}} = \frac{(\dim \lambda)^2}{|\lambda|!} \frac{ \theta^{2|\lambda|}}{|\lambda|!} e^{-\theta^2}, \] which is now a probability measure on $\bY$ (the {\bf Poissonized Plancherel measure}). If we fix $|\lambda|$, the conditional distribution is $\frac{(\dim \lambda)^2}{|\lambda|!}$. \begin{remark} The Robinson--Schensted algorithm gives a bijection $\{\sigma \in S_n\} \leftrightarrow \{(P,Q)\}$ where $(P,Q)$ is a pair of standard Young tableaux of the same shape with $n$ boxes. One property of this algorithm is that the length of the longest increasing (decreasing) subsequence in $\sigma$ (interpreted as a sequence $\sigma(1), \cdots, \sigma(n)$) become the first row (column) of the shape. %Using this algorithm, the uniform measure on $S_n$ goes to $(\dim %\lambda)^2 / n!$. %We can also consider the Poisson distribution on the quarter plane. %This is related to polynuclear growth %11/3/11 Consider a Poisson process in $\bR_+^2$. Take a square with a vertices $(0,0)$, $(0,\theta)$, $(\theta,0)$, and $(\theta, \theta)$. Then the probability of having $n$ points in this square is $\frac{\theta^{2n}}{n!} e^{-\theta^2}$. If we rotate the picture 45 degrees to the left, we can interpret the Poisson points as (multilayer) polynuclear growth: at each point, it gives rise to a cone with rays parallel to the $x$- and $y$-axes. We think of time as going straight up and space as going left to right (in the rotated picture). When two cones hit each other, the rays stop and give rise to a new cone of a different ``color'' (note that the colors have an ordering). For each color, we can ask how many rays of that given color that separate $(0,0)$ and $(\theta, \theta)$. This gives a partition whose size is the number of Poisson points in the square. In fact, if we interpret the Poisson points as a permutation (using their relative ordering with respect to $x$- and $y$-coordinates) this partition is the one obtained via the Robinson--Schensted algorithm. \steven{This is also known as Viennot's shadow diagram construction, see Section 3.6 of Bruce Sagan's book.} \end{remark} \begin{example}[$z$-measures] Fix $z,z' \in \bC$ and $\xi \in (0,1)$. Define $\pi_x(p_k) = z \xi^{k/2}$ and $\pi_y(p_k) = z' \xi^{k/2}$. Positivity of the resulting specialization corresponds to the condition $z' = \ol{z}$. In particular, we have \[ \frac{\pi_x(s_\lambda) \pi_y(s_\lambda)}{\sum_\lambda \pi_x(s_\lambda) \pi_y(s_\lambda)} = \frac{(\dim \lambda)^2}{(|\lambda|!)^2} \prod_{b \in \lambda} (z+c(b))(z'+c(b)) \xi^{|\lambda|} (1-\xi)^{zz'} \] where $c$ is the content of the box $b \in \lambda$. This is related to Robinson--Schensted--Knuth: given a rectangular grid, let a box $(i,j)$ have value $\omega_{ij} = k$ with probability $(1-\xi)\xi^k$. Note that $\max_P \sum_{(i,j) \in P} \omega_{ij} = \lambda_1$ where the maximum is over all up-right paths $P$ from the bottom-left box to the top-right box of the rectangle, and $\lambda$ is the partition coming from RSK. The $z$-measure can be used to study the asymptotics. In particular, the probability of getting a shape $\lambda$ is given by the $z$-measure of $\lambda$ with $z$ and $z'$ specialized to the dimensions of the grid. One could instead study measures on paths of the form $z^{-1} \exp(\beta \sum_{(i,j) \in P} \omega_{ij})$. For large $\beta$ this is as above, but for small $\beta$, the problem requires new ideas to analyze. \end{example} Now we will show that Schur measures are determinantal processes. Given a partition $\lambda$, we will associate the point configuration $\ul{X}(\lambda) = \{\lambda_i - i + \frac{1}{2} \mid i \ge 0\} \subset \bZ + \frac{1}{2}$. Such subsets contain all but finitely many negative values, so we can think of it as a perturbation of a densely packed point configuration. Given such a configuration, we get a finite point configuration by applying particle-hole involution to the negative part. The resulting point configuration gives the modified Frobenius coordinates of $\lambda$: the positive particles describe the rows and the negative particles describe the columns. The correlation functions are \[ \rho_n(x_1, \dots, x_n) = \Prob \{ \lambda \in \bY \mid \ul{X}(\lambda) \supset \{x_1, \dots, x_n\}\}. \] \begin{theorem} \label{thm:schurmeasuredet} For $u_1, \dots, u_n \in \bZ + \frac12$, we have \[ \rho_n(u_1, \dots, u_n) = \det(K(u_i,u_j))_{i,j=1}^n, \] where $K(i,j)$ is the coefficient of $v^iw^{-j}$ in the formal series \[ \frac{H_x(v)H_y(w^{-1})}{H_y(v^{-1}) H_x(w)} \sum_{k \ge 0} (wv^{-1})^{k+\frac12} \] where $H_x(z) = \sum_{m \ge 0} \pi_x(h_m) z^m$. \end{theorem} \begin{corollary} \label{cor:schurmeasuredet} In the analytic setting, \[ K(i,j) = \frac{1}{(2\pi \sqrt{-1})^2} \int_{C_v} \int_{C_w} \frac{H_x(v) H_y(w^{-1})}{H_y(v^{-1}) H_x(w)} \frac{\sqrt{vw}}{v-w} \frac{ {\rm d}v\, {\rm d} w}{v^{i+1}w^{-j+1}}, \] where $C_v$ is the positively oriented unit circle $|v|=1$ and $C_w$ is the positively oriented circle $|w| = 1-\eps$ with $1 \gg \eps$. Then $\sum \pi_x(h_m)z^m$ being holomorphic in a neighborhood of a unit disc implies that $\sum_\lambda \pi_x(s_\lambda) \pi_y(s_\lambda) < \infty$. \end{corollary} Strong Szeg\"o theorem: given $\{a_n\}_{n \in \bZ}$, set $D_M = \det(a_{i-j})_{i,j=1}^M$ and $A(z) = \sum_n a_n z^n$. What is the asymptotics of $D_M$? \begin{proposition}[Gessel] For any $n=1,2,\dots$, we have \[ \sum_{\substack{\lambda \in \bY \\ \lambda_1 \le n}} s_\lambda(x) s_\lambda(y) = \det(f_{i-j})^n_{i,j=1} \] where $\sum_{m \in \bZ} f_m v^m = E_x(v^{-1})E_y(v)$ and $E_x(z) = \sum_{m \ge 0} e_m(x) z^m$. \end{proposition} \begin{proof} The dual Jacobi--Trudi formula gives $s_\lambda = \det(e_{\ell_i - j + 1})_{i,j=1}^n$ where $\ell_i = \lambda'_i + n - i$. Then \begin{align*} \sum_{\lambda_1 \le n} s_\lambda(x) s_\lambda(y) = \sum_{\ell_1 > \cdots > \ell_n \ge 0} \det(e_{\ell_i - j + 1}(x)) \det(e_{\ell_i - j + 1}(y)) = \det(F_{i,j}), \end{align*} where $F_{i,j} = \sum_{m \in \bZ} e_{m-i}(x) e_{m-j}(y) = f_{i-j}$ by Cauchy--Binet. \end{proof} \begin{corollary} $\det(f_{i-j})_{i,j=1}^n = Z \det(1-K)_{\ell^2(n+1, n+2, \dots)}$ and $Z = \exp(\sum_k p_k(x) p_k(y)/k)$. \end{corollary} Note that the determinant tends to 1 as $n$ gets large. Take $\pi_x(f) = f(x_1, \dots, x_N)$ and $\pi_y(f) = f(y_1, \dots, f_N)$. Then \[ s_\lambda(x_1, \dots, x_N) = \frac{\det(x_i^{\lambda_j + N - j})_{i,j=1}^N}{\det (x_i^{N-j})_{i,j=1}^N} = \frac{\det(x_i^{\ell_j})_{i,j=1}^N}{\rm const.} \] where $\ell_i = \lambda_i + N - i$. Note that \[ s_\lambda(x) s_\lambda(y) = {\rm const.} \det(x_i^{\ell_j}) \det(y_i^{\ell_j}) = \det(\phi_i(\ell_j)) \det(\psi_i(\ell_j)). \] %11/8/11 \begin{example} The Poissonized Plancherel measure is \[ H({\rm Pl}_\theta; u) = \sum_{n \ge 0} {\rm Pl}_\theta(h_n) u^n = \sum_{n \ge 0} \frac{(\theta u)^n}{n!} = e^{\theta u}. \qedhere \] \end{example} \begin{lemma} \label{lemma:cauchyinverse} Let $G_{ij} = (1-x_iy_j)^{-1}$ be the entries of the Cauchy matrix. Then \[ [G^{-T}]_{k,\ell} = \frac{\prod_{j=1}^N (1-x_jy_\ell) (1-x_ky_j)}{ (1-x_ky_\ell) \prod_{j=1, j \ne k}^N (x_k - x_j) \prod_{j=1, j \ne \ell}^N (y_\ell - y_j)}. \] \end{lemma} \begin{proof} First note that \[ \det(G) = \det\left(\frac{1}{1-x_iy_j}\right)_{i,j=1}^N = \frac{\prod_{i N$. So the measure lives on the set of partitions with at most $N$ parts. In this case, we can write $s_\lambda(x_1, \dots, x_N) = {\rm const}(x) \det(x_i^{\ell_j})_{i,j=1}^N$ where the constant is independent of $\lambda$ and $\ell_j = \lambda_j - j + N$. Similarly, we have $s_\lambda(y_1, \dots, y_N) = {\rm const}(y)\det(y_i^{\ell_j})_{i,j=1}^N$, and so \[ \Prob\{\lambda\} = {\rm const.} \det(x_i^{\ell_j})_{i,j=1}^N \det(y_i^{\ell_j})_{i,j=1}^N. \] If we set $\phi_i(\ell) = x_i^\ell$ and $\psi_j(\ell) = y_j^\ell$, we can realize this as a biorthogonal ensemble. The correlation kernel in this case is \[ K(\ell, \ell') = \sum_{i,j=1}^N G_{ij}^{-T} \phi_i(\ell) \psi_j(\ell') \] where \[ G_{ij} = \langle \phi_i, \psi_j \rangle_{\ell^2(\bZ_+)} = \sum_{\ell \ge 0} x_i^\ell y_j^\ell = \frac{1}{1-x_iy_j}. \] Hence by Lemma~\ref{lemma:cauchyinverse}, we have \[ K(a,b) = \sum_{k,\ell=1}^N \frac{\prod_{j=1}^N (1-x_jy_\ell) (1-x_ky_j)}{(1-x_ky_\ell) \prod_{j \ne k} (x_k-x_j) \prod_{j \ne \ell}(y_\ell - y_j)} x_k^a y_\ell^b. \] Using the residue formula, we can rewrite this as \[ \frac{1}{(2\pi\sqrt{-1})^2} \int_{C_w} \int_{C_z} \prod_{k=1}^N \frac{(1-wy_k)(1-zx_k)}{(w-x_k)(z-y_k)} \frac{w^az^b}{1-zw} {\rm d}z\, {\rm d}w \] where $C_w$ goes around $\{x_k\}$ and $C_z$ goes around $\{y_\ell\}$. Note that $1-zw$ is not a pole in this path integral. Note that $H(u) = \prod_{k=1}^N (1-x_ju)^{-1}$ which gives the formula in Corollary~\ref{cor:schurmeasuredet}. \end{proof} \begin{proof}[Proof 2 of Theorem~\ref{thm:schurmeasuredet}] Recall from Giambelli's formula that for $\lambda = (p_1, \dots, p_d | q_1, \dots, q_d)$, we have \begin{align*} \Prob\{\lambda\} &= \frac{s_\lambda(x) s_\lambda(y)}{Z} = \frac{\det(s_{p_i | q_j}(x))_{i,j=1}^d \det(s_{p_i|q_j}(y))_{i,j=1}^d}{Z}. \end{align*} Define two $\bZ_+ \times \bZ_+$ matrices $A_{pq} = s_{p|q}(x)$ and $B_{pq} = s_{p|q}(y)$ and set $L = \begin{pmatrix} 0 & B \\ -A^T & 0 \end{pmatrix}$, which is an operator in $\ell^2(\bZ_+) \oplus \ell^2(\bZ_+) = \ell^2(\bZ_+ \coprod \bZ_+)$. The Schur measure is the $L$-ensemble with this $L$. To compute the correlation kernel, one needs to compute $L(1+L)^{-1}$. Note that \begin{align*} \sum_{p,q \ge 0} A_{pq} u^p v^q = \sum_{p,q \ge 0} s_{p|q} u^p v^q = \frac{1}{u+v} \left( \frac{H(u)}{H(-v)} - 1\right) \end{align*} where $H(u) = \sum_n h_nu^n$. For notation, we set $\phi_+(u) = \phi_-(-u^{-1}) = H(u)$. Define \[ \Phi(z,w) = \frac{\phi_-(z) \phi_+(w^{-1})}{\phi_+(z) \phi_-(w^{-1})}. \] Then we have $L(1+L)^{-1} = \begin{pmatrix} K_{11} & K_{12} \\ K_{21} & K_{22} \end{pmatrix}$ where \begin{align*} (K_{11})_{pq} &= \frac{1}{(2\pi\sqrt{-1})^2} \int_{|zw| < 1} \frac{\Phi(z,w) z^pw^q}{1-zw} {\rm d} z\, {\rm d} w\\ (K_{12})_{pq} &= \frac{(-1)^q}{(2\pi\sqrt{-1})^2} \int_{|zw| < 1} \frac{\Phi(z,w) w^{p}}{(1-zw)z^{q+1}} {\rm d} z\, {\rm d} w\\ (K_{21})_{pq} &= \frac{(-1)^{p+1}}{(2\pi\sqrt{-1})^2} \int_{|zw| < 1} \frac{\Phi(z,w) w^q}{(1-zw)z^{p+1}} {\rm d} z\, {\rm d} w\\ (K_{22})_{pq} &= \frac{(-1)^{p+q}}{(2\pi\sqrt{-1})^2} \int_{|zw| < 1} \frac{\Phi(z,w)}{(1-zw)z^{p+1}w^{q+1}} {\rm d} z\, {\rm d} w. \qedhere \end{align*} \end{proof} \subsubsection{Fock space approach.} This approach is due to Okounkov. Set $\bZ' = \bZ + \frac{1}{2}$. Let $V$ be a linear space with basis $\{\ul{k} \mid k \in \bZ' \}$. The linear space $\bigwedge^{\infty/2} V$ (space of semi-infinite forms) has a basis $v_S = \ul{s_1} \wedge \ul{s_2} \wedge \ul{s_3} \wedge \cdots$ with $S = \{s_1 > s_2 > \cdots\} \subset \bZ'$ such that $S_+ = S \setminus \bZ'_{< 0}$ and $S_- = \bZ'_{<0} \setminus S$ are both finite. Note that if $|S_+| = |S_-|$, then $S = \{\lambda_i - i + \frac{1}{2}\}_{i \ge 1}$ for a partition $\lambda$. Let $C$ be the {\bf charge operator} defined by $Cv_S = (|S_+| - |S_-|)v_S$ (so $\ker C$ is identified with $\bY$). The empty partition corresponds to $\ul{-\frac12} \wedge \ul{-\frac32} \wedge \cdots$, and is called the vacuum vector $v_{\rm vac} = v_\emptyset$. For each $k \in \bZ'$, the creation (fermion) operators are defined by $\psi_k(f) = \ul{k} \wedge f$ and $\psi_k^*$ is the adjoint of $\psi_k$ with respect to the dot product $\langle, \rangle$ that makes $\{v_S\}$ orthonormal. These operators satisfy the relation \[ \{ \psi_k, \psi_\ell^* \} = \psi_k \psi_\ell^* + \psi_\ell^* \psi_k = \delta_{k,\ell} \] for $k, \ell \in \bZ'$. Note that \[ \psi_k \psi_k^* v_S = \begin{cases} 0 & \text{if } k \notin S \\ v_S & \text{if } k \in S \end{cases}. \] The Boson operators are given by $\alpha_n = \sum_{k \in \bZ'} \psi_{k-n} \psi_k^*$ for $n \in \bZ \setminus \{0\}$. They satisfy the Heisenberg relations \[ [\alpha_m,\alpha_n] = \alpha_m \alpha_n - \alpha_n \alpha_m = m \delta_{m,n} \in \bZ \setminus \{0\}. \] Introduce (vertex) operators \[ \Gamma_{\pm}(p_1, p_2, \dots) = \exp(\sum_{n \ge 1} \frac{p_n}{n} \alpha_{\pm n}). \] Some basic properties: \begin{compactenum} \item $\Gamma_{\pm}^* = \Gamma_{\mp}$. \item $\DS \Gamma_+(p_1, p_2, \dots) \Gamma_-(p_1', p_2', \dots) = \exp(\sum_{n \ge 1} \frac{p_n p'_n}{n}) \Gamma_-(p'_1, p'_2, \dots) \Gamma_+(p_1, p_2, \dots)$. \item $\Gamma_+ v_{\rm vac} = v_{\rm vac}$ and $\Gamma_-(p_1, p_2, \dots) v_{\rm vac} = \sum_{\lambda \in \bY} s_\lambda v_{\ul{X}(\lambda)}$, where $s_\lambda$ is the polynomial in the $p_i$ that comes from the Murnaghan--Nakayama rule. \end{compactenum} Using this notation, we have \[ \rho_n(x_1, \dots, x_n) = {\rm const.} \langle (\prod_{i=1}^n \psi_{x_i} \psi^*_{x_i}) \Gamma_-(p_1, p_2, \dots) v_{\rm vac}, \Gamma_-(p'_1, p'_2, \dots) v_{\rm vac} \rangle \] Using the relations, we can rewrite this as \[ \rho_n(x_1, \dots, x_n) = \langle \prod_{i=1}^n \Psi_{x_i} \Psi_{x_i}^* v_{\rm vac}, v_{\rm vac} \rangle = \det(K(x_i, x_j))_{i,j=1}^n \] where \[ \Psi_k G \psi_k G^{-1}, \quad \Psi_k^* = G \psi_k^* G^{-1}, \quad G = \Gamma_+(p_1, p_2, \dots) \Gamma_-^{-1}(p'_1, p'_2, \dots), \quad K(x,y) = \langle \Psi_x \Psi_y^* v_{\rm vac}, v_{\rm vac} \rangle. \] This $K(x,y)$ can be interpreted in terms of the contour integral. %11/10/11 \subsubsection{Asymptotics of the (Poissonized) Plancherel measures.} We consider the Plancherel measure \[ \Prob\{\lambda\} = e^{-\theta^2} \frac{\theta^{2|\lambda|}}{|\lambda|!} \frac{(\dim \lambda)^2}{|\lambda|!} \] where $\theta > 0$. Choose $n$ according to ${\rm Pois}(\theta^2)$, choose a uniformly random permutation $\sigma \in S_n$, do RSK, and then read off the shape. The longest increasing subsequence of the random permutation is $\lambda_1$. Let $\ell_n$ be the longest increasing subsequence for $S_n$ and $\ell_{\theta^2}$ be the longest increasing subsequence for $S_n$ where $n \sim {\rm Pois}(\theta^2)$. Ulam's problem (1961): What is $\ell_n$ on average for $n \gg 1$? He conjectured that $\bE \ell_n \sim \sqrt{n}$. Airplane boarding problem: suppose there is an airplane with one aisle and one seat per row, and $n$ passengers need to board (and each passenger is assigned a seat in advance). Also suppose that each passenger takes $t$ units of time to seat. Thinking of the boarding order as a passenger, the length of the longest increasing subsequence gives how long it takes to board everyone. Hammersley (1972): The limit $\lim_{n \to \infty} \ell_n / \sqrt{n}$ exists. Vershik--Kerov, Logan--Shepp (1977): This limit is 2. Baik--Deift--Johansson (1999): $\lim_{n \to \infty} \{ (\ell_n - 2\sqrt{n})/n^{1/6} \le s \} = F_2(s)$ Consider Young diagrams of size $n$ scaled by $1/\sqrt{n}$. Vershik--Kerov proved that there is a curve $C$ such that for any $\eps > 0$, the boundary of Young diagrams fit into the $\eps$ neighborhood of $C$ with probability 1. This curve is given by \[ \Omega(u) = \begin{cases} \frac{2}{\pi}(u \arcsin(u/2) + \sqrt{4-u^2}) & |u| \le 2 \\ |u| & |u| > 2 \end{cases}. \] Note that \[ \Omega'(u) = \frac{2}{\pi} \arcsin(u/2) \quad (\text{for } |u| < 2). \] \begin{theorem} \label{thm:bulk} Take $X^{(n)}_1, X^{(n)}_2, \dots, X^{(n)}_k \in \bZ'$ such that \[ \lim_{n \to \infty} \frac{X^{(n)}_j}{\sqrt{n}} = \cK_{\rm bulk}(x|u) = \frac{\sin (\phi x)}{\pi x} \in (-2,2) \] where $\phi = \arccos(u/2)$, $x \in \bZ$, $n \sim {\rm Pois}(\theta^2)$, and $X_i - X_j$ is independent of $n$. Then \[ \rho_k(X_1, \dots, X_k) = \Prob(\ul{X}(\lambda) \supset \{X_1, \dots, X_k\}) = \det(\cK_{\rm bulk}(X_i - X_j|u)_{i,j=1}^k) \] \end{theorem} \begin{example} For $k=1$, get $\rho_1(X) \to \frac{1}{\pi} \arccos(u/2)$ which is the global density of $\ul{X}(\lambda)$. And we have $\rho_1 = (1-\Omega')/2$. For $0 < c < \pi$, the (translation invariant) random determinantal point process on $\bZ$ with correlation kernel $K(x,y) = \frac{\sin(c(x-y))}{x-y}$ is called the {\bf discrete sine process}. Then $\rho_1 = c/\pi$, so the average number of particles in a window of size $L$ is $\sim Lc/\pi$, and the variance is $\sim \ln L$. \end{example} For two groups $X_1, \dots, X_k$, and $Y_1, \dots, Y_\ell$ with $X_j / \sqrt{n} \to u$ and $Y_j / \sqrt{n} \to v$, we have \[ \rho_{k+\ell}(X_1, \dots, X_k, Y_1, \dots, Y_\ell) \sim \rho_k(X_1, \dots, X_k) \rho_\ell(Y_1, \dots, Y_\ell). \] \begin{proof}[Proof of Theorem~\ref{thm:bulk}] The correlation kernel is \[ K(x,y; \theta) = \frac{1}{(2\pi\sqrt{-1})^2} \int_{|z| = c_z > c_w = |w|} \frac{\exp(\theta(z-z^{-1} - w + w^{-1}))}{z-w} \frac{{\rm d}z\, {\rm d}w}{z^{x+\frac{1}{2}} w^{-y+\frac{1}{2}}}. \] We can also write \[ K(x,y; \theta) = \theta\frac{J_{x-\frac12} J_{y+\frac12} - J_{x+\frac12} J_{y-\frac12}}{x-y} \] where $\sum_{n \in \bZ} J_n z^n = \exp(\theta(z-z^{-1}))$ ($J_n$ are the Bessel functions). Consider the case $x/ \theta \to u$ and $y/\theta \to u$ for $u \in (-2,2)$. Then we have \[ K(x,y) = \frac{1}{(2\pi\sqrt{-1})^2} \int \frac{\exp(\theta( z-z^{-1} - \frac{x}{\theta}\ln z ))}{\exp( \theta(w-w^{-1} - \frac{y}{\theta} \ln w))} \frac{{\rm d}z\, {\rm d}w}{(z-w)\sqrt{zw}}. \] Define $F(z) = z-z^{-1} - u\ln z$. Then $F(z) \in \sqrt{-1}\bR$ for $|z|=1$. We want to find a contour on which $|e^{\theta F(z)}|$ is small, i.e., want ${\rm Re}\, F(z) < 0$. Note that $zF'(z) = z+z^{-1} - u$. So $F'(e^{i\phi})=0$ is equivalent to $2\cos \phi = u$. So we consider the gradient of ${\rm Re}\, F(z)$ on the unit circle and deform the unit circle to get a $z$-contour where ${\rm Re}\, F(z) < 0$. For the $w$-contour we deform in the opposite direction. But we do not want them to intersect. So first we deform $w$-contour, take the integral out, and then deform the $z$-contour. Then the limit above converges to \[ \frac{1}{2\pi i} \int_{e^{-i \arccos(u/2)}}^{e^{i\arccos(u/2)}} \frac{{\rm d} w}{w^{x-y+1}} = \frac{\sin(\phi(x-y))}{\pi(x-y)}. \qedhere \] \end{proof} What about the case $u = \pm 2$ (edge case)? Then $F'(1) = F''(1) = 0$. So the two critical points $e^{\pm i \arccos(u/2)}$ merge and form the ``Monkey saddle''. We have $F(z) \sim \frac13 (z-1)^3 + O((z-1)^4)$ near $z=1$. So near the critical point, we have $\exp(\frac{\theta}{3} (z-1)^3)$. Introduce local coordinates $z= 1 + iz'\theta^{-1/3}$ and $w = 1 + iw'\theta^{-1/3}$. Then $\theta F(z) = -\theta \frac{i}{3}(z')^3 + \text{lower order terms}$. Write $x = 2\theta + \tilde{x} \theta^{1/3}$. Then $\tilde{x} \ln z = \tilde{x} \theta^{1/3} \ln(1 + iz' \theta^{-1/3}) \sim i \tilde{x} z'$. The conclusion is that \[ K(2\theta + \tilde{x} \theta^{1/3}, 2\theta + \tilde{y} \theta^{1/3}) \sim \frac{\theta^{-1/3}}{(2\pi)^2} \int \frac{\exp(-i(z')^3/3 - iz'x)}{\exp(-i(w')^3/3 - iw'y)} \frac{{\rm d} z' \, {\rm d} w'}{i(z'-w')}. \] %11/15/11 We call this limit the {\bf Airy kernel} and denote it by $K_{\rm Airy}(x,y)$. Use the identity \[ \exp(-iz'x + iw'y) = \frac{i}{x-y} (\frac{\partial}{\partial z'} + \frac{\partial}{\partial w'}) \exp(-iz'x + iw'y) \] to write \[ K_{\rm Airy}(x,y) = \frac{{\rm Ai}(x) {\rm Ai}'(y) - {\rm Ai}(y) {\rm Ai}'(x)}{x-y} \] where \begin{align*} {\rm Ai}(x) &= \frac{1}{2\pi} \int \exp(-i(z')^3/3 - iz'x) {\rm d}z'\\ &= \frac{1}{\pi} \int_0^\infty \cos(xs + s^3/3){\rm d}s. \end{align*} This is the unique (up to constant) solution of $A''(x) = xA(x)$ decaying at $+\infty$. This is called the {\bf Airy function}. For $\ell = 2\theta + s\theta^{1/3}$, we have \begin{align*} \Prob\{\lambda_1 - \frac{1}{2} \le \ell \} &= \det(1-K)|_{L^2(\ell+1, \ell+2, \dots)}\\ &= 1 - \sum_{x \ge \ell+1} K(x,x) + \frac{1}{2!} \sum_{x,y\ge \ell+1} \det \begin{pmatrix} K(x,x) & K(x,y) \\ K(y,x) & K(y,y) \end{pmatrix} - \cdots\\ &\approx 1 - \int_{x \ge s} K_{\rm Airy}(s,s) {\rm d}s + \frac{1}{2!} \int_{x,y \ge s} \det \begin{pmatrix} K_{\rm Airy}(x,x) & K_{\rm Airy}(x,y) \\ K_{\rm Airy}(y,x) & K_{\rm Airy}(y,y) \end{pmatrix} - \cdots\\ &= \det(1-K_{\rm Airy})_{L^2(s,+\infty)}. \end{align*} Hence \[ \lim_{\theta \to \infty} \Prob\left\{ \frac{\text{longest increasing subsequence} - 2\theta}{\theta^{1/3}} \le s \right\} = \det(1-K_{\rm Airy})_{L^2(s,+\infty)} =: F_2(s). \] This is known as the Tracy--Widom GUE distribution. This comes up in random matrix theory as follows: under GUE($N$), we have \[ \Prob\left\{ \frac{\text{max. eigenvalue}(H) - \sqrt{2N}}{N^{-1/6}} \le s \right\} = F_2(s). \] Tracy--Widom characterized the function $F_2(s)$ as being a solution to the ordinary differential equation $u''(s) = s u(s) + 2u(s)^3$ (Painlev\'e II) where $u(s) = -(\ln F_2(s))''$. So $u(s) \sim {\rm Ai}(s)$ for $s\to +\infty$. We could consider the problem of studying the joint distribution of the longest increasing sequences going to two different points $(\theta_1, \eta_1)$ and $(\theta_2, \eta_2)$ in a collection of Poisson distributed points in the first quadrant of the plane. This is unknown if the two points lie on a common ray from the origin. \subsection{Measures given by products of determinants.} Recall that biorthogonal ensembles are given by \[ c \det (\Phi_i(x_j))_{i,j=1}^N \det(\Psi_i(x_j))_{i,j=1}^N. \] \begin{example} Take $U \in U(N)$ Haar-distributed. Then $\Spec(U) = \{u_1, \dots, u_N\} \subset T$ and \[ P({\rm d} u) = c \prod_{i j \end{cases}\\ G_{i,j} &= \Phi_i * T_{1,k} * \Psi_j. \end{align*} and define \[ K(x^{(p)}, y^{(q)}) = -\delta_{q < p} T_{q,p}(y^{(q)}, x^{(p)}) + \sum_{i,j=1}^N (G^{-T})_{i,j}(\Phi_i * T_{1,p})(x^{(p)}) (T_{q,k} * \Psi_j)(y^{(q)}). \] \begin{theorem}[Eynard--Mehta] \label{thm:eynardmehta} This process is determinantal with correlation kernel $K(x^{(p)}, y^{(q)})$. \end{theorem} Biorthogonal ensembles: kernel simplifies if $G = 1$ (orthogonal polynomial ensembles). Algebraically ``nice'' case (realizable for GUE($N$)): there exists an orthonormal basis $\{\Xi_i^{(j)}\}_{i \ge 1}$ in each $L^2(\cX^{(j)})$ such that \[ T_{j,j+1}(x,y) = \sum_{i \ge 1} c_{j,j+1;i} \Xi^{(j)}_i(x) \Xi_i^{(j+1)}(y) \] (this means that $T_{j,j+1} * \Xi^{(j+1)}_m = c_{j,j+1;m} \Xi^{(j)}_m$) and such that \begin{align*} {\rm span}\{\Xi^{(1)}_1, \dots, \Xi^{(1)}_N\} &= {\rm span}\{\Phi_i\}_{i=1}^N\\ {\rm span}\{\Xi^{(k)}_1, \dots, \Xi^{(k)}_N\} &= {\rm span}\{\Psi_i\}_{i=1}^N \end{align*} (so in the formula for $W(X)$, we can replace the $\Phi$ and $\Psi$ by $\Xi^{(1)}$ and $\Xi^{(k)}$). Then \[ K(x^{(p)}, y^{(q)}) = \begin{cases} \sum_{i=1}^N c_{p,q;i}^{-1} \Xi_i^{(p)}(x) \Xi_i^{(q)} (y) & p \le q \\ -\sum_{i > N} c_{q,p;i} \Xi^{(p)}_i(x) \Xi^{(q)}_j(y) & p > q \end{cases} \] where $c_{p,q;i} = c_{p,p+1;i} c_{p+1,p+2;i} \cdots c_{q-1,q;i}$. \subsubsection{Natural source of such processes.} Consider a (finite) directed acyclic graph with vertex set $V$ and edge set $E$. Let $w \colon E \to \bC$ be an arbitrary weight function. For any path $\pi$ in the graph, denote $w(\pi) = \prod_{e \in \pi} w(e)$. For several paths, the weight of the collection is defined to be the product of the weights of the individual paths. For $u,v \in V$, denote by $\Pi(u,v)$ the set of all directed paths $u \to v$, and define $T(u,v) = \sum_{\pi \in \Pi(u,v)} w(\pi)$. For any two paths, their intersection is any edges or vertices they share in common. \begin{theorem} Let $(u_1, \dots, u_n)$ and $(v_1, \dots, v_n)$ be any $n$-tuples of vertices. Assume that for any non-identical permutation $\sigma \in S_n$, we have $\{(\pi_1, \dots, \pi_n) \mid \pi_i \in \Pi(u_i, v_{\sigma(i)}), \pi_i \cap \pi_j = \emptyset\} = \emptyset$. Then \[ \sum_{\substack{\pi_j \in \Pi(u_j,v_j)\\ \pi_i \cap \pi_j \ne \emptyset} } w(\pi_1, \dots, \pi_n) = \det(T(u_i, v_j))_{i,j=1}^n. \] \end{theorem} We can use this to express the probability that collections of paths go from $u_1 ,\dots, u_n$ to $v_1, \dots, v_n$, and have specified points along their paths, as a product of determinants of the form given by the weight function $W(X)$ given above. Say we consider paths between vertices $u_1, u_2$ and $v_1, v_2$. Then \begin{align*} W(\text{pairs of paths not intersecting}) &= W(\text{all pairs}) - W(\text{pairs intersect}).\\ &= T(u_1,v_1) T(u_2, v_2) - T(u_1, v_2) T(u_2, v_1). \end{align*} There is a continuous time analog (Karlin--McGregor 1951). Consider a stationary stochastic process whose state space is an interval of the extended real line $\bR \cup \{\pm \infty\}$. Assume that the process has the {\it strong Markov property} and that its paths are continuous everywhere. Take $n$ points $x_1 < \cdots < x_n$, and $n$ Borel sets $E_1 < \cdots < E_n$ ($E_i < E_{i+1}$ means $E_i$ is completely to the left of $E_{i+1}$). Suppose that $n$ particles start from $x_1, \dots, x_n$, and execute the process simultaneously and independently. Then \[ \det(P_t(x_i, E_j))_{i,j=1}^n \] is the probability that at time $t$, the particles will be in $E_1, \dots, E_n$, respectively, without any of them coinciding along the way, where $P_t$ is the transition probability of the Markov process. %11/17/11 \begin{lemma} \label{lemma:2x2inverse} Let $A,B,C,D$ be matrices. Then \[ \begin{pmatrix} A & B \\ C & D \end{pmatrix}^{-1} = \begin{pmatrix} -M^{-1} & M^{-1} B D^{-1} \\ D^{-1} C M^{-1} & D^{-1} - D^{-1} C M^{-1} B D^{-1} \end{pmatrix} \] where $M = BD^{-1} C - A$. \end{lemma} \begin{proof} The right hand side is $\begin{pmatrix} 1 & 0 \\ -D^{-1} C & 1 \end{pmatrix} \begin{pmatrix} -M^{-1} & M^{-1} B D \\ 0 & D^{-1} \end{pmatrix}$, so its inverse is $\begin{pmatrix} -M & B \\ 0 & D \end{pmatrix} \begin{pmatrix} 1 & 0 \\ DC & 1 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:eynardmehta}] We realize the point process as a conditional $L$-ensemble. Let the state space be $\tilde{\cX} = \{1, \dots, N\} \cup \cX$ and set $\Prob\{X\} = {\rm const.} \det L\begin{pmatrix} 1 & \cdots & N & X \\ 1 & \cdots & N & X \end{pmatrix}$ where $L$ is the block matrix \[ L = \begin{pmatrix} & \Phi \\ & & -T_{1,2} \\ & & & -T_{2,3} \\ & & & & -T_{3,4} & \cdots \\ \vdots \\ \Psi \end{pmatrix} \] where the blocks are $\{1,\dots, N\}, \cX^{(1)}, \cX^{(2)}, \dots, \cX^{(k)}$, and $\Phi_{i,x^{(1)}} = \Phi_i(x^{(1)})$, $\Psi_{x^{(k)}, j} = \Psi_j(x^{(k)})$, and $(T_{j,j+1})_{x^{(j)}, x^{(j+1)}} = T_{j,j+1}(x^{(j)}, x^{(j+1)})$. The correlation kernel for this $L$-ensemble is given by $K_{\cX \times \cX} = 1_{\cX} - (1_{\cX} + L)^{-1}|_{\cX \times \cX}$. Using Lemma~\ref{lemma:2x2inverse} with $A = 0$, $B = \begin{pmatrix} \Phi & 0 & \cdots & 0 \end{pmatrix}$, $C = \begin{pmatrix} 0 & \cdots & 0 & \Psi \end{pmatrix}^T$ and $D = \begin{pmatrix} 1 - T_{1,2} & \cdots & 0 \\ \vdots \\ 0 & \cdots & 1 - T_{k-1,k} \\ 0 & \cdots & 1 \end{pmatrix}$. Then \begin{align*} D^{-1} &= \begin{pmatrix} 1-T_{1,2} & \cdots & 0 \\ \vdots \\ 0 & \cdots & 1-T_{k-1,k} \\ 0 & \cdots & 1 \end{pmatrix}^{-1} = (1-\begin{pmatrix} T_{1,2} & \cdots & 0 \\ \vdots \\ 0 & \cdots & T_{k-1,k} \\ 0 & \cdots & 0 \end{pmatrix})^{-1}\\ &= 1 + \begin{pmatrix} T_{1,2} & \cdots & 0 \\ \vdots \\ 0 & \cdots & T_{k-1,k} \\ 0 & \cdots & 0 \end{pmatrix} + \begin{pmatrix} T_{1,2} & \cdots & 0 \\ \vdots \\ 0 & \cdots & T_{k-1,k} \\ 0 & \cdots & 1 \end{pmatrix}^2+ \cdots\\ &= \begin{pmatrix} 1 & T_{1,2} & T_{1,3} & \cdots & T_{1,k} \\ 0 & 1 & T_{2,3} & \cdots & T_{2,k} \\ \vdots \\ 0 & \cdots & 1 \end{pmatrix}. \end{align*} Then we find that $M = \Phi T_{1,k} \Psi =: G$ with $G_{i,j} = \Phi_i * T_{1,k} * \Psi_j$ for $i,j=1,\dots, N$. So we have \begin{align*} K = 1-(\begin{pmatrix} 1 & T_{1,2} & T_{1,3} & \cdots & T_{1,k} \\ 0 & 1 & T_{2,3} & \cdots & T_{2,k} \\ \vdots \\ 0 & \cdots & 1 \end{pmatrix} - \begin{pmatrix} 1 & T_{1,2} & T_{1,3} & \cdots & T_{1,k} \\ 0 & 1 & T_{2,3} & \cdots & T_{2,k} \\ \vdots \\ 0 & \cdots & 1 \end{pmatrix} \begin{pmatrix} 0 \\ \vdots \\ 0 \\ \Psi \end{pmatrix}\\ G^{-1} \begin{pmatrix} \Phi & 0 & \cdots & 0 \end{pmatrix} \begin{pmatrix} 1 & T_{1,2} & T_{1,3} & \cdots & T_{1,k} \\ 0 & 1 & T_{2,3} & \cdots & T_{2,k} \\ \vdots \\ 0 & \cdots & 1 \end{pmatrix}) \end{align*} This is equivalent to \[ K(x^{(p)}, y^{(q)}) = -T_{p,q}(x^{(p)}, y^{(q)}) + \sum_{i,j=1}^N G_{ij}^{-T} (\Phi_i * T_{i,p})(x^{(p)}) (T_{q,k} * \Psi_j)(y^{(q)}). \qedhere \] \end{proof} A modification: Consider a path in the Gelfand--Tsetlin graph $\lambda^{(1)} \prec \lambda^{(2)} \prec \cdots \prec \lambda^{(N)}$ with $\lambda^{(k)} = (\lambda^{(k)}_1 \ge \dots \ge \lambda_k^{(k)})$ and recall that $\mu \prec \lambda$ if and only if $\lambda_1 \ge \mu_1 \ge \lambda_2 \ge \mu_2 \ge \cdots$. This interlacing condition is satisfied when $\lambda$ is the spectrum of an $N \times N$ real symmetric (or Hermitian) matrix, and $\mu$ is the spectrum of its upper left $(N-1) \times (N-1)$ submatrix. Set $\cX = \cX_1 \cup \cdots \cup \cX_N$ and pick $X \in \Conf(\cX)$ with $X$ having exactly $j$ points in each $\cX_j$. Let $x_N^{(N-1)} = \cdots = x_2^{(1)} = x_1^{(0)}$ be a ``virtual particle''. Now set \begin{align*} W(X) &= \det(\Psi_i(x_j^{(N)}))_{i,j=1}^N \det(\phi_N(x_i^{(N-1)}, x_j^{(N)}))_{i,j=1}^N\\ &\quad\quad \det(\phi_{N-1}(x_i^{(N-2)}, x_j^{(N-1)}))_{i,j=1}^{N-1} \cdots \det(\phi_1(x_1^{(0)}, x_1^{(1)}))_{i,j=1}^1 \end{align*} We want to prove that this is a determinantal process and compute the correlation kernel. Set \[ \phi^{(n_1, n_2)} = \begin{cases} \phi_{n_1 + 1} * \phi_{n_1+2} * \cdots * \phi_{n_2} & n_1 < n_2 \\ 0 & n_1 \ge n_2 \end{cases}. \] Define the Gram matrix \[ G_{k,\ell} = \phi_k * \phi^{(k,n)} * \Psi_\ell \] for $k,\ell = 1,\dots,N$. \begin{theorem} \[ K(n_1, x_1, n_2, x_2) = -\phi^{(n_2,n_1)}(x_2,x_1) + \sum_{i=1}^{n_1} \sum_{j=1}^N [G^{-T}]_{i,j} (\phi_i * \phi^{(i,n_1)})({\rm virt.}, x_1) (\phi^{(n,N)} \Psi_j)(x_2) \] \end{theorem} \begin{proof} This is similar to the proof from before, we just give the definition of $L$: \[ L = \begin{pmatrix} 0 & \psi_1 & \psi_2 & \cdots \\ \\ & & -\phi^{(1,2)} \\ & & & -\phi^{(2,3)} & \cdots \\ \vdots \\ \Psi \end{pmatrix} \] where $\psi_i$ is a matrix which is 0 except for the $i$th row, which is given by $\phi_i({\rm virt.}, x)$. \end{proof} Determinantal point processes also appear in the following contexts: \begin{compactenum} \item Random dimer models. Start from a finite bipartite graph. Look at perfect matchings of this graph with the uniform distribution. Then dimers (edges) form a determinantal point process. The kernel is the inverse of the Kasteleyn matrix. If we assume that the bipartite graph is a hexagonal lattice, then we can consider the dual graph, and the dimers in a perfect matching tell us how to glue together the triangles in the dual graph to get rhombi. \item The zeroes of a Gaussian analytic function $\sum_{n \ge 0} a_n z^n$ in the unit disk form a determinantal process. \item If we renormalize the nontrivial zeroes of the Riemann zeta function $\zeta(s) = \sum_{n \ge 1} n^{-s}$, they should form a determinantal process. \end{compactenum} Recall that the extreme Gibbs measures on the Gelfand--Tsetlin graph are parametrized by certain sequences $\{\alpha_i^{\pm}, \beta_i^{\pm}, \gamma_i^{\pm}\}$. Also, we have \[ \Prob\{\lambda_1 \ge \cdots \ge \lambda_N \} = \dim_N \lambda = \prod_{i < j} \frac{ \lambda_i - i - (\lambda_j - j)}{j-i} \] and \[ \sum_{n \in \bZ} \phi_n z^n = \exp(\gamma^+(z-1) + \gamma^-(z^{-1} - 1)) \prod_i \frac{(1 + \beta_i^+(z-1))(1+\beta^-_i(z^{-1}-1))}{(1 - \alpha_i^+(z-1))( 1 - \alpha_i^-(z^{-1}-1))}. \] We write $\lambda_j^{(m)} - j = x^{(m)}_j$ for $j=1,\dots,m$ and $m=1,\dots,n$. Then \[ \Prob \{\lambda^{(1)} \prec \cdots \prec \lambda^{(N)}\} = \det(\phi(\lambda^{(N)}_i - i + j) )_{i,j=1}^N = \det(\phi(x_i^{(N)} + j)_{i,j=1}^N. \] We want to encode the interlacing condition on the path by determinants via \[ \Prob\{(x_j^{(m)})\} = \det(\phi(x_i^{(N)} + j)) \det(x^{(N)}, x^{(N-1)}) \cdots \det(x^{(2)}, x^{(1)}) \] where $\det(x^{(i)}, x^{(i-1)})$ is 0 if $\lambda^{(i)}$ and $\lambda^{(i-1)}$ don't interlace, and is 1 otherwise. %11/22/11 Let $\{F_n\}_{n \in \bZ}$ be a sequence and set $F(z) = \sum_{n \in \bZ} F_n z^n$. Assume that this is holomorphic in an annulus $A_r = \{z \in \bC \mid r < z < r^{-1} \}$ for some $r \in (0,1)$. Pick $N \ge 1$ and $a_1, \dots, A_n \in A_r$ such that $F(a_j) \ne 0$. Given this data, assign to any path $\lambda^{(1)} \prec \cdots \prec \lambda^{(N)}$ in $\bG\bT$ the weight \[ W(\lambda^{(1)}, \dots, \lambda^{(N)}) = \det(F_{\lambda^{(N)}_i - i + j})_{i,j=1}^N \prod_{k=1}^N a_k^{|\lambda^{(k)}| - |\lambda^{(k-1)}|}. \] In the special case that $a_k = 1$ for all $k$, we get \[ F(u) = \exp(\gamma^+(u-1) + \gamma^-(u^{-1} - 1)) \prod_i \frac{(1 + \beta_i^+(u-1)) (1+\beta_i^-(u^{-1} - 1))}{ (1+\alpha_i^+(u-1)) (1-\alpha_i^-(u^- - 1))}, \] which corresponds to a point in $\partial \bG \bT$. So the measure on paths that we get is one of the extremal Gibbs measures. It turns out that it is easier to prove this statement for general $a_k$ and let them go to 1. The {\bf partition function} is \[ z(a_1, \dots, a_N) = \sum_{\lambda^{(1)} \prec \cdots \prec \lambda^{(N)}} W(\lambda^{(1)}, \dots, \lambda^{(N)}) = F(a_1) \cdots F(a_N) \] (the second equality will be proved later). We define \[ \Prob(\lambda^{(1)} \prec \cdots \prec \lambda^{(N)}) = \frac{W(\lambda^{(1)}, \dots, \lambda^{(N)})}{z(a_1, \dots, a_N)} \] and to any path $\lambda^{(1)} \prec \cdots \prec \lambda^{(N)}$, we assign a point configuration \[ \cL(\lambda^{(1)} \prec \cdots \prec \lambda^{(N)}) = \{ (n, \lambda_j^{(n)} - j) \mid n=1,\dots,N,\ j=1,\dots,n\}. \] \begin{theorem} The point process on $[N] \times \bZ$ generated by $\cL$ is determinantal with correlation kernel \begin{align*} K(n_1, x_1; n_2, x_2) &= -\delta_{n_1 < n_2} \frac{1}{2\pi i} \int_{\Gamma_0} \frac{u^{x_1 - x_2 - 1}}{\prod_{j=n_1 + 1}^{n_2} (1-a_j u)} {\rm d} u \\ &\quad\quad+ \frac{1}{(2\pi i)^2} \int_{u \in \Gamma_0} \int_{w \in \Gamma_{a^{-1}}} \frac{F(u^{-1})}{F(w^{-1})} \frac{u^{x_1}}{w^{1 + x_2}} \frac{\prod_{j=1}^{n_1} (1-a_j u)}{ \prod_{k=1}^{n_2} (1-a_kw)} \frac{{\rm d}u\ {\rm d}w}{u-w} \end{align*} where $\Gamma_0$ is a contour in $A_r$ around 0 not containing any $a_k$ and $\Gamma_{a^{-1}}$ is a contour containing the poles $w = a_j^{-1}$ and no other singularities. \end{theorem} \begin{proof} We'll use the Eynard--Mehta theorem. For $n \ge 1$, define $\phi_n \colon \bZ \times \bZ \to \bC$ by \[ \phi_n(x,y) = \begin{cases} (a_n)^{x-y} & x \le y \\ 0 & x > y \end{cases}. \] Also for $x \in \bZ$ define $\phi_n({\rm virt.}, x) = a_n^x$. \begin{lemma} For all $n \ge 1$ and $x_1^{n-1} > x_2^{n-1} > \cdots > x_{n-1}^{n-1}$ and $x_1^n > \cdots > x_n^n \in \bZ$, we have \[ \det(\phi_n(x^{n-1}_k, x^n_\ell))_{k,\ell=1}^n = \begin{cases} a_n^{|x^n| - |x^{n-1}|} & x_1^n \ge x_1^{n-1} > x_2^n \ge x_2^{n-1} > \cdots \\ 0 & \mathrm{else} \end{cases}. \] \end{lemma} \begin{proof} It suffices to handle the case where $a_k = 1$ for all $k$ since we can multiply each row by $(a_n)^{-x_k^{n-1}}$ and each column by $a_n^{x_\ell^n}$. The interlacing condition implies that either the matrix is upper triangular, or we get 2 rows or 2 columns are the same. \end{proof} Set \[ \Psi_j(x) = \frac{1}{2\pi i} \int_{|u|=1} F(u) p_j(u^{-1}) \frac{{\rm d}u}{u^{x+2}} \] where $p_j(z) = \prod_{i \ne j} (1-a_iz)$, a polynomial of degree $N-1$. Then \[ \det (\Psi_j(x_k)) = \frac{1}{(2\pi i)^N} \int_{\substack{u_1, \dots, u_N\\u_1^2 + \cdots + u_N^2 = 1}} F(u_1) \cdots F(u_N) \det(p_j(u_k^{-1}))_{j,k=1}^N \frac{{\rm d}u_1 \cdots {\rm d}u_N}{u_1^{x_1 + 2} \cdots u_N^{x_N+2}}. \] This inner determinant is a Vandermonde matrix which can be replaced by $\det(u_k^{-j+1})_{k,j=1}^N$ using row operations. So the above expression simplifies to \[ \det \left( \frac{1}{2\pi i} \int F(u_k) u_k^{-1} \frac{{\rm d}u_k}{u_k^{x_k + j + 1}} \right). \] The Gram matrix is \[ G_{k,\ell} = \phi_k * \phi_{k+1} * \cdots * \phi_N * \Psi_\ell. \] Then \[ \phi_n(x,y) = \frac{1}{2\pi i} \int \frac{1}{1-a_n u} \frac{{\rm d} u}{u^{y-x+1}} = \begin{cases} a_n^{y-x} & y \ge x \\ 0 & {\rm else} \end{cases}. \] Then we get \[ G_{k, \ell} = \prod_{j=k+1}^N \phi_j(a_k^{-1}) F(a_k) p_\ell(a_k^{-1}) a_k^{-1}, \] and so $G$ is a diagonal matrix and hence \[ (G^{-T})_{k,\ell} = \delta_{k,\ell} a_k F(a_k)^{-1} \prod_{j=1}^{k-1} \phi_j(a_k^{-1}). \qedhere \] \end{proof} %11/29/11 \section{Toeplitz-like transition probabilities.} \begin{proposition} Choose $\alpha_1, \dots, \alpha_n \in \bC^*$ and let $F(z)$ be an analytic function in an annulus centered at $0$ that contains each $\alpha_i^{-1}$. Also assume that $F(\alpha_j^{-1}) \ne 0$. Then \[ \frac{1}{F(\alpha_1^{-1}) \cdots F(\alpha_n^{-1})} \sum_{\{y_1 < \cdots < y_n \} \subset \bZ} \det (\alpha_i^{y_j})_{i,j=1}^n \det( f(x_i -y_j))_{i,j=1}^n = \det(\alpha_i^{x_j})_{i,j=1}^n \] where $f(n) = \frac{1}{2\pi i} \int F(z) \frac{{\rm d}z}{z^{n+1}}$, i.e., $F(z) = \sum_{n \in \bZ} f(n) z^n$. \end{proposition} \begin{proof} We have \begin{align*} \sum_{y_1 < \cdots < y_n} \det(\alpha_i^{y_j}) \det(f(x_i - y_j)) % &= \frac{1}{n!} \sum_{y_1, \dots, y_n \in \bZ} \det(\alpha_i^{y_j}) % \det(f(x_i - y_j))\\ = \det(\sum_{y \in \bZ} \alpha_k^y f(x_j - y))_{k,j=1}^n \end{align*} by Cauchy--Binet. Also, \begin{align*} \sum_{y \in \bZ} \alpha_k^y f(x_j - y) &= \sum_{y \in \bZ} \frac{1}{2\pi i} \int \frac{\alpha_k^y F(z) {\rm d}z}{z^{x_j-y+1}}\\ &= \frac{1}{2\pi i} \int_{|z| < |\alpha_k^{-1}|} F(z) \sum_{y \ge x_k + 1} \frac{\alpha_k^y {\rm d}z}{z^{x_j-y+1}} + \frac{1}{2\pi i} \int_{|z| > |\alpha_k^{-1}|} F(z) \sum_{y \le x_k} \frac{\alpha_k^y {\rm d}z}{z^{x_j-y+1}}\\ &= \frac{1}{2\pi i} \int_{|z| < |\alpha_k^{-1}|} F(z) {\rm d} z \frac{\alpha_k^{x_j + 1}}{1 - \alpha_k z} - \frac{1}{2\pi i} \int_{|z| > |\alpha_k^{-1}|} F(z) {\rm d} z \frac{\alpha_k^{x_j + 1}}{1 - \alpha_k z}\\ &= \operatorname{Res}_{z = \alpha_k^{-1}} \frac{F(z) \alpha_k^{x_j + 1}}{1- \alpha_k z} = \alpha_k^{x_j} F(\alpha_k^{-1}) \qedhere \end{align*} \end{proof} This says that \begin{align} \label{eqn:poissonpathprob} \text{``Prob''}\{(x_1 < \cdots < x_n) \to (y_1 < \cdots < y_n)\} = \frac{1}{F(\alpha_1^{-1}) \cdots F(\alpha_n^{-1})} \frac{\det (\alpha_i^{y_j})}{\det (\alpha_i^{x_j})} \det(f(x_i - y_j)). \end{align} \begin{example} A {\bf Poisson path} is the graph of a simple continuous time random walk. Say that we have $n$ Poisson processes starting at $(0,0), (0,1), \dots, (0,n-1)$ in $\bZ^2$ conditioned not to intersect ending at heights $X_1, \dots, X_n$ at time $T$. We want to let $T \to \infty$ and let the $X_i \to \infty$ in such a way that the finite time pictures have a meaningful limit. To be able to do this, we need that $X_j / T \to a_j$ where $0 < a_1 \le a_2 \le \cdots \le a_n < \infty$. This gives a measure on paths. The resulting limit can be seen as a Markov process on $\{x_1 < \cdots < x_n\}$ started from $\{0 < 1 < \cdots < n-1\}$ with transition probabilities \[ P_t((x_1 < \cdots < x_n) \to (y_1 < \cdots < y_n)) = {\rm const.} \det\left( \frac{e^{-t} t^{y_i - x_j}}{(y_i - x_j)!}\right)_{i,j=1}^n \frac{\det(a_i^{y_j})}{\det(a_i^{x_j})}. \] The last fraction is called {\bf Doob's $h$-transform}. Also note that the right hand side is the expression in \eqref{eqn:poissonpathprob} with $F(z) = e^{tz^{-1}}$. \end{example} \begin{proposition} Assume $y_n$ is virtual with $f(x_k - {\rm virt.}) = \alpha_n^{x_k}$. Then \[ \frac{1}{F(\alpha_1^{-1}) \cdots F(\alpha_{n-1}^{-1})} = \sum_{y_1 < \cdots < y_{n-1}} \det (\alpha_i^{y_j})_{i,j=1}^{n-1} \det (f(x_i - y_j))_{i,j=1}^n = \det (\alpha_i^{x_j})_{i,j=1}^n. \] \end{proposition} \begin{proof} The proof is the same as before once we expand the second determinant along the last column. \end{proof} Set $X = (x_1 < \cdots < x_n)$ and $Y = (y_1 < \cdots < y_n)$. Define the {\bf Toeplitz matrix with symbol $F$} as $(f(i-j))_{i,j \in \bZ}$. Also define \begin{align*} T_n(\alpha_1, \dots, \alpha_n; F)(X \to Y) &= \frac{1}{F(\alpha_1^{-1}) \cdots F(\alpha_n^{-1})} \frac{\det (\alpha_i^{y_j})_{i,j=1}^n}{\det (\alpha_i^{x_j})_{i,j=1}^n} \det(f(x_i - y_j)_{i,j=1}^n)\\ T_{n-1}^n(\alpha_1, \dots, \alpha_n; F)(X \to Y) &= \frac{1}{F(\alpha_1^{-1}) \cdots F(\alpha_{n-1}^{-1})} \frac{\det (\alpha_i^{y_j})_{i,j=1}^{n-1}}{\det (\alpha_i^{x_j})_{i,j=1}^n } \det(f(x_i - y_j)_{i,j=1}^n). \end{align*} \begin{proposition} Let $F_1, F_2$ be two holomorphic functions in an annulus with $F_j(\alpha_k^{-1}) \ne 0$. Then \begin{align*} T_n(F_1) T_n(F_2) &= T_n(F_2) T_n(F_1) = T_n(F_1 F_2)\\ T_n(F_1) T_{n-1}^n(F_2) &= T_{n-1}^n(F_1) T_{n-1}(F_2) = T_{n-1}^n (F_1F_2). \end{align*} \end{proposition} \begin{proof} First note that \begin{align*} T_n(F_1)T_n(F_2)(X,Y) &= \sum_{z_1 < \cdots < z_n} \frac{\det(\alpha_i^{z_j})}{\det (\alpha_i^{x_j})} \det (f_1(z_j - x_i)) \frac{\det(\alpha_i^{y_j})}{\det (\alpha_i^{z_j})} \det (f_2(y_i - z_j)). \end{align*} The rest follows from the fact that the Fourier transform turns multiplication into convolution. \end{proof} \begin{lemma} For $F(z) = 1 + pz$, then for $x_1 < \cdots < x_n$ and $y_1 < \cdots < y_n$, we have \[ \det(f(x_i - y_j))_{i,j=1}^n = \begin{cases} p^{\sum_{i=1}^n (x_i - y_i)} & \mathrm{if\ } y_i - x_i \in \{-1,0\} \mathrm{\ for\ } 1 \le i \le n\\ 0 & \mathrm{else}\end{cases}. \] \end{lemma} \begin{lemma} For $F(z) = (1-qz)^{-1}$, then for $x_1 < \cdots < x_n$ and $y_1 < \cdots < y_n$, we have \[ \det(f(x_i - y_j)) = \begin{cases} q^{\sum_{i=1}^n (x_i - y_i)} & \mathrm{if\ } x_{i-1} < y_i \le x_i \mathrm{\ for\ } 1 \le i \le n \\ 0 & \mathrm{else} \end{cases}. \] \end{lemma} \begin{lemma} For $F(z) = p + qz(1-qz)^{-1}$, then for $x_1 < \cdots < x_n$ and $y_1 < \cdots < y_n$, we have \[ \det(f(x_i - y_j)) = \begin{cases} q^{\sum_{i=1}^n (x_i - y_i)} p^{\#\{i \mid x_i = y_i\}} (1-p)^{\#\{i \mid x_{i-1} = y_i\}} & \mathrm{if\ } x_{i-1} \le y_i \le x_i \mathrm{\ for\ } 1 \le i \le n\\ 0 & \mathrm{else} \end{cases}. \] \end{lemma} There is a way to say all of these things in the language of Schur functions. The reason is the Jacobi--Trudi formula $s_{\lambda/\mu} = \det(h_{\lambda_i - \mu_j - i + j})_{i,j=1}^N$ where $N \ge \ell(\lambda)$. For arbitrary specializations $y,z,t \colon \Lambda \to \bC$, we can define two probabilities. The first is \[ p_{\lambda, \mu}^{\uparrow}(y,z) = \frac{1}{H(y,z)} \frac{s_\mu(y)}{s_\lambda(y)} s_{\mu / \lambda}(z) \] where $H(\rho_1, \rho_2) = \sum_{\lambda \in \bY} s_\lambda(\rho_1) s_\lambda(\rho_2)$ (cf. the Doob $h$-transform). To check that this is a probability (i.e., $\sum_\mu p_{\lambda, \mu}^{\uparrow}(y,z) = 1$), we use the identities \begin{align*} \sum_\lambda s_{\lambda / \nu} (\rho_1) s_{\lambda / \hat{\nu}}(\rho_2) &= H(\rho_1, \rho_2) \sum_\mu s_{\nu / \mu}(\rho_2) s_{\hat{\nu} / \mu}(\rho_1)\\ \sum_\nu s_{\kappa/\nu}(\rho_1) s_{\nu / \hat{\kappa}}(\rho_2) &= s_{\kappa / \hat{\kappa}}(\rho_1, \rho_2). \end{align*} The second probability is \[ \rho_{\lambda, \nu}^{\downarrow}(y;t) = \frac{s_\nu(y)}{s_\lambda(y,t)} s_{\lambda / \nu}(t). \] We can similarly check that $\sum_\nu \rho_{\lambda, \nu}^{\downarrow}(y;t) = 1$. \begin{proposition} Let $y,z,\dots$ be nonnegative specializations of $\Lambda$ (i.e., take nonnegative values on $s_\lambda$). Then \begin{align*} \sum_\mu p_{\lambda, \mu}^\uparrow(y;z_1) p_{\mu, \nu}^\uparrow(y;z_2) &= \sum_{\hat{\mu}} p_{\lambda, \hat{\mu}}^\uparrow(y;z_2) p_{\hat{\mu}, \nu}^\uparrow(y;z_1)\\ p^\downarrow(y,t_2;t_1) p^\downarrow(y; t_2) &= p^\downarrow(y, t_1; t_2) p^\downarrow(y; t_1)\\ p^\uparrow(y,t;z) p^\downarrow(y; t) &= p^\downarrow(y, t) p^\uparrow(y; z)\\ \end{align*} \end{proposition} Here we interpret $p^\downarrow(\rho_1, \rho_2; \rho_3)$ by combining $\rho_1$ and $\rho_2$ into one specialization (this can be defined by setting $p_k(\rho_1, \rho_2) = p_k(\rho_1) + p_k(\rho_2)$ for the power sum symmetric functions). \begin{proof} For the third identity, we have \begin{align*} \sum_\mu p^\uparrow_{\lambda, \mu}(y,t;z) p^\downarrow_{\mu, \nu}(y;t) &= \frac{1}{H(y,t;z)} \sum_\mu \frac{s_\mu(y,t)}{s_\lambda(y,t)} s_{\mu / \lambda}(z) \frac{s_\nu(y)}{s_\mu(y,t)} s_{\mu/\nu}(t)\\ &= \frac{1}{H(y,t;z)} \sum_\kappa s_{\lambda / \kappa}(t) \frac{s_\kappa(y)}{s_\lambda(y,t)} s_{\nu / \kappa}(z) \frac{s_\nu(y)}{s_\kappa(y)}\\ &= \sum_\kappa p^\downarrow_{\lambda, \kappa}(y,t) p^\uparrow_{\kappa, \nu}(y;z). \qedhere \end{align*} \end{proof} \begin{remark} Let $\bS(x,y)$ be the Schur measure on $\bY$ given by $\Prob(\lambda) = s_\lambda(x) s_\lambda(y) / H(x,y)$. Then \begin{align*} \bS(x,y) p^\uparrow (y;z) &= \bS(x,z;y)\\ \bS(x;y,t) p^\downarrow(y;t) &= \bS(x;y) \qedhere \end{align*} \end{remark} %12/1/11 \begin{example} Let $y$ be the specialization into $N$ variables $a_1, \dots, a_N > 0$. Then \[ p^\uparrow_{\lambda,\mu}(y,z) = \frac{1}{H(y;z)} \frac{\det(a_i^{m_j})}{\det(a_i^{N-j})} \frac{\det(a_i^{N-j})}{\det (a_i^{\ell_j})} s_{\mu / \lambda}(z) = \frac{1}{H(y;z)} \frac{\det(a_i^{m_j})}{\det (a_i^{\ell_j})} \det(h_{m_i - \ell_j}) \] where $m_i = N + \mu_i - i$ and $\ell_j = N + \lambda_j - j$. \end{example} Let $S$ and $S^*$ be some countable sets. Assume we have stochastic matrices $\Lambda \colon S^* \times S \to [0,1]$ (so $\sum_b \Lambda(a,b) = 1$) and $P^* \colon S^* \times S^* \to [0,1]$ and $P \colon S \times S \to [0,1]$. We will assume that $P^* \Lambda = \Lambda P$, and we define this product to be $\Delta$. We can visualize this as a diagram \[ \xymatrix{ S^* \ar[r]^-{P^*} \ar[d]^-{\Lambda} & S^* \ar[d]^-{\Lambda} \\ S \ar[r]^-P & S }. \] We want a bivariate Markov operator $P^{(2)} \colon ((x^*,x) \to (y^*,y))$ that would project to both $P$ and $P^*$. One possibility, due to Diaconis--Fill (1990): Set $S^{(2)} = \{(x^*,x) \in S^* \times S \mid \Lambda(x^*,x) > 0 \}$. Define \[ P^{(2)} \colon ((x^*,x) \to (y^*,y)) = P(x \to y) \frac{P^*(x^*,y^*) \Lambda(y^*, y)}{\Delta(x^*,y)} \] Then $P^{(2)}$ is stochastic: \[ \sum_{(y^*,y) \in S^* \times S} P^{(2)}((x^*,x) \to (y^*,y)) = \sum_{y \in S} \sum_{y^* \in S^*} P^{(2)}((x^*,x) \to (y^*,y)) = \sum_{y \in S} P(x,y) = 1. \] \begin{definition} $\Prob\{(x^*,x)\} = \Lambda(x^*,x)$ is a {\bf $\Lambda$-Gibbs measure} on $S^* \times S$ with top coordinate fixed at $x^*$. A general $\Lambda$-Gibbs measure has the form $\Prob(x^*, x) = m^*(x^*) \Lambda(x^*,x)$ where $m^*$ is a probability measure on $S^*$. \end{definition} \begin{lemma} \begin{align*} \sum_{y^* \in S^*} P^{(2)}((x^*,x) \to (y^*,y)) &= P(x,y)\\ \sum_{x \in S} \Lambda(x^*, x) P^{(2)}((x^*,x) \to (y^*,y)) &= P^*(x^*,y^*) \Lambda(y^*,y) \end{align*} In particular, the projection of $P^{(2)}$ to $S$ is $P$ and the projection of $P^{(2)}$ to Gibbs measures on $S^* \times S$ is $P^*$. \end{lemma} \begin{proof} The first identity follows by definition. For the second identity, we have \[ \sum_{x \in S} \Lambda(x^*, x) P(x,y) \frac{P^*(x^*,y^*) \Lambda(y^*, y)}{\Delta(x^*,y)} = P^*(x^*,y^*) \Lambda(y^*,y). \qedhere \] \end{proof} Now set $S^* = \{x_1 < \cdots < x_n\} \subset \bZ^n$ and $S = \{x_1 < \cdots < x_{n-1}\} \subset \bZ^{n-1}$. Also set $P^* = T_n(\alpha_1, \dots, \alpha_n; 1 + \beta z)$ and $P = T_n(\alpha_1, \dots, \alpha_{n-1}; 1 + \beta z)$, so that \begin{align*} P^*(x \to y) &= {\rm const.} \frac{\det(\alpha_i^{y_j})}{ \det(\alpha_i^{x_j})} \beta^{\sum_j (x_j - y_j)}\\ P(x \to z) &= \frac{\det(\alpha_i^{z_j})_{i,j=1}^{N-1}}{ \det(\alpha_i^{x_j})_{i,j=1}^N} \alpha_n^{\sum_j (x_j - z_j)}. \end{align*} Finally, define $\Lambda = T^n_{n-1}(\alpha_1, \dots, \alpha_n; (1-\alpha_n z)^{-1})$. What is $P^*(x^*, y^*) \Lambda(y^*, y) / \Delta(x^*,y)$ as a probability distribution on the $y^*$ (with $x^*, y$ fixed)? We get \[ C_1 \frac{\det(\alpha_i^{y_j^*})}{ \det(\alpha_i^{x_j^*})} \beta^{|x^*| - |y^*|} \frac{\det(\alpha_i^{y_j})}{ \det(\alpha_i^{y_j^*})} \alpha_n^{|y| - |y^*|} = C_2 (\alpha_n \beta)^{|x^*| - |y^*|} \] where $C_1, C_2$ are constants. The conditions $P^*(x^*,y^*) \ne 0$ and $\Lambda(y^*, y) \ne 0$ enforce interlacing conditions. For each $x_j^*$ to obtain $y_j^*$ there are 3 possibilities: \begin{compactenum} \item $x_j^*$ is free to move by one using a Bernoulli random variable with probability of success equal to $\alpha_n \beta$, \item $x_j^*$ must move, \item $x_j^*$ must stay. \end{compactenum} In conclusion, $P^* \Lambda / \Delta$ is a product of independent Bernoulli random variables. We can also give an $n$-level construction. Pick sets $\fX_1, \dots, \fX_n$ and assume we are given $\Lambda_{k-1}^k \colon \fX_k \times \fX_{k-1} \to [0,1]$. Also assume that on every level there is a stochastic matrix $P_k \colon \fX_k \times \fX_k \to [0,1]$ and that we have commutativity relations $\Lambda_{k-1}^k P_{k-1} = P_k \Lambda_{k-1}^k$ and let $\Delta_{k-1}$ be this product. Define an $n$-variate Markov chain on $\fX^{(n)} = \{(x_n, \dots, x_1) \in \fX_n \times \cdots \times \fX_1 \mid \Lambda_{k-1}^k(x_k, x_{k-1}) \ne 0\}$ by \[ P^{(n)} \left( \begin{pmatrix} x_n \\ \vdots \\ x_1 \end{pmatrix} \to \begin{pmatrix} y_n \\ \vdots \\ y_1 \end{pmatrix} \right) = P_1(x_1 \to y_1) \frac{P_2(x_2 \to y_2) \Lambda_1^2(y_2, y_1) }{\Delta_1(x_2, y_1)} \cdots \frac{P_n(x_n \to y_n) \Lambda_{n-1}^n(y_n, y_{n-1})}{\Delta_{n-1}(x_n, y_{n-1})}. \] The projection to the bottom $k$ levels gives the $k$-variate chain on $P^{(k)}$. The projection to Gibbs measures $\cM^{(N)}(x_1, \dots, x_n) = m_n(x_n) \Lambda_{n-1}^n(x_n, x_{n-1}) \cdots \Lambda_1^2(x_2,x_1)$ gives Gibbs measure with new $\tilde{m_n} = m_nP_n$. Here the Gibbs property of a measure on $\fX_N$ is: given $x_n$, the conditional distribution of $(x_1, \dots, x_{N-1})$ is $\Lambda_{N-1}^N(x_N,x_{N-1}) \cdots \Lambda_1^2(x_2, x_1)$. The proofs of these facts are similar to the bivariate case. Now set $P_k = T_k(\alpha_1, \dots, \alpha_k; 1 + \beta z^{-1})$ and $\Lambda_{k-1}^k = T_{k-1}^k(\alpha_1, \dots, \alpha_{k-1}; (1-\alpha_k z^{-1})^{-1})$. Then $\Prob(x_1 \text{ jumps by } 1) = \alpha_1 \beta (1+\alpha_1 \beta)^{-1}$ and $\Prob(x_1 \text{ stays}) = (1+\alpha_1\beta)^{-1}$. Also, $\Prob(x_2 \text{ jumps by } 1) = \alpha_2 \beta (1+\alpha_2 \beta)^{-1}$ and $\Prob(x_2 \text{ stays}) = (1+\alpha_2\beta)^{-1}$, etc. %12/6/11 Now start with a Gibbs measure $m^{(N)}$ as above. Define \[ m^{(N)}_t = m^{(N)} (P^{(N)})^t \] for $t=0,1,2,\dots$. We want to know joint distributions for different $t$. \begin{compactitem} \item $m_t^{(N)}(x_1, \dots, x_N) = (m_N P_N^t) \Lambda_{N-1}^N(x_N,x_{N-1}) \cdots \Lambda_1^2(x_2,x_1)$ \item For any time moments $t_N \le t_{N-1} \le \cdots \le t_1$, we have \begin{align*} \Prob(x_N(t_N), x_{N-1}(t_{N-1}), \dots, x_1(t_1)) &= m_N P_N^{t_N}(x_N) \Lambda_{N-1}^N P_{N-1}^{t_N - t_{N-1}}(x_N, x_{N-1})\\ &\quad \quad \Lambda_{N-2}^{N-1} P_{N-2}^{t_{N-1} - t_{N-2}}(x_{N-1}, x_{N-2}) \cdots \Lambda_1^2 P_1^{t_1 - t_2}(x_2, x_1). \end{align*} \end{compactitem} The most general marginal that can be computed using $P_1, \dots, P_N$ allows us to specify $x_N, \dots, x_1$ at many time moments, i.e., $\Prob(x_N(t_N^1), \dots, x_N(t_N^{s_N}), x_{N-1}(t_{N-1}^1), \dots)$. \begin{example} \[ \xymatrix{ \bG\bT^+_N \ar[d] \ar[r] & \bG\bT_N^+ \ar[d] \\ \bG\bT^+_{N-1} \ar[d] \ar[r] & \bG\bT_{N-1}^+ \ar[d] \\ \vdots \ar[d] & \vdots \ar[d] \\ \bG\bT^+_1 \ar[r] & \bG\bT_1^+} \] For parameters $a_1, \dots, a_N > 0$ (assume $a_j \equiv 1$), we have \begin{align*} P_k(\lambda \to \mu) &= {\rm const.} \frac{s_\mu(a_1, \dots, a_k)}{s_\lambda(a_1, \dots, a_k)} s_{\mu / \lambda}(\rho)\\ \Lambda_{k-1}^k(\nu \to \kappa) &= \frac{s_\kappa(a_1, \dots, a_{k-1})}{ s_\nu(a_1, \dots, a_k)} s_{\nu / \kappa}(a_k) \end{align*} where \[ \sum_{n \ge 0} h_n(\rho) z^n = e^{\gamma z} \prod_{i \ge 1} \frac{1 + \beta_i z}{1 - \alpha_i z}. \qedhere \] \end{example} \begin{proposition} If one starts with the Gibbs measure with ($\rho_0$ arbitrary) \[ m_N(\lambda^{(N)}) = \frac{1}{z} s_\lambda(a_1, \dots, a_N) s_\lambda(\rho_0), \] then applying $P^{(N)}$ with data in the previous example gives the Gibbs measure with \[ m_N(\lambda^{(N)}) = \frac{1}{z} s_\lambda(a_1, \dots, a_N) s_\lambda(\rho_0, \rho_1). \] \end{proposition} \begin{example} \begin{compactitem} \item Take $\rho$ to be such that $\sum_n h_n(\rho) z^n = 1 + \beta z$. At level $i$, there are $i$ particles. Particles can be pushed by lower ones or be blocked by lower ones. Each particle jumps at most 1 to the right. The particle jumps with probability $\beta a_i / (1+ \beta a_i)$. The possible starting and ending points of each particle is forced by the interlacing conditions. \item Assume instead that $\sum_n h_n(\rho) z^n = (1-\alpha z)^{-1}$. The jumping is similar, but Bernoulli jumps are replaced by geometrically distributed jumps: particle at level $i$ can now jump $k$ steps with probability $(a_i \alpha)^k (1 - a_i \alpha)$. Again the possible starting and ending points of each particle is forced by the interlacing conditions. \end{compactitem} Now restrict attention to the smallest coordinates on each level. Consider the first example with $a_j \equiv 1$. On level $i$, the right most particle jumps with probability $\beta/(1+\beta)$, and then we consider the next one, etc. This is called the discrete time totally asymmetric simple exclusion process with sequential update. \end{example} What happens if $\beta \to 0$ or $\alpha \to 0$ in these two examples? Then the Markov operators converge to a generator of a continuous time Markov process, i.e., $\lim_{\beta \to 0} (P^{(N)}(\beta))^{\tau / \beta} = \Pi_\tau$ where $P^{(N)}(\beta) = 1 + \tau \Pi^{(N)} + \cdots$. What is the generator of the continuous time Markov process? (Note that $\lim_{\beta \to 0} (1 + \beta z)^{\tau / \beta} e^{\tau z}$.) Each particle tries to jump to the right by 1 independent of the others with exponential waiting time. It can be blocked by a lower particle, and it can also be pushed by higher particles. On the line of leftmost particles at each level, we observe a continuous time process on $\bZ$ (no pushing). This is totally asymmetric simple exclusion process (TASEP), which was introduced by Spitzer (1970). On the line of rightmost particles, we see the long range TASEP. What are equilibrium distributions for TASEP? \begin{theorem} Translation-invariant distributions with fixed density are products of Bernoullis, i.e., the probability of a particle occupying a each given site is $p$, independent of the other sites. \end{theorem} The non-trivial aspect is that there is a parameter. This parameter is the average density of the particles. Random growth model: take a point configuration in $\bZ$. Above each particle put a line segment of slope $-1$, otherwise put a line segment of slope $+1$. Match the endpoints of adjacent sites. If a particle jumps to the right, this corresponds to a change from $\vee$ to $\wedge$. One can think of this as adding a box on top of the line segments. So one can think of the dynamics as adding a box at any permissible location independently with rate 1. Suppose that this line segment is approximating a smooth curve $h(x,t)$. What happens in the limit? This is the hydrodynamic limit of the system, which ends up being deterministic. If the local density is $\rho$, then the local slope is $\partial_x h = (1-\rho)/2$. Pick a space between sites; how often does it get crossed by a particle? For a crossing, would need to have a particle to the left and no particle to the right. The probability of this setup is $\partial_t h = \rho(1-\rho)$. So $h$ is a solution of the differential equation \[ \frac{\partial p}{\partial t} = \frac{\partial}{\partial x}(p (1-\rho)), \] which is the inviscid Bugen's equation, or shock equation. %11/8/11 We have partitions $\lambda^{(1)} \prec \cdots \prec \lambda^{(N)}$ which we identify with the points $\{\lambda^{(k)}_j - j\}_{k \ge 1, 1 \le j \le k}$. On $\lambda^{(k)}$ at time $t$ we have a Schur measure (right jumps allowed) or more generally the measure on $\bG\bT$ patterns is an extreme Gibbs measure on $\bG\bT$. The process generated by $\{(k, \lambda_j^{(k)}- j)\} \subset \bZ^2$ is determinantal. On level $n$ and at time $t$, in coordinates $\{k+\lambda_j^{(k)} - j\}_{k \ge 1, 1 \le j \le k}$ the correlation kernel is \[ K(x_1,x_2) = \frac{1}{(2\pi i)^2} \int_{\Gamma_1} {\rm d}z \int_{\Gamma_{0,z}} \frac{z^n}{e^{tz}(1-z)^{x_1 + 1}} \frac{e^{tw} (1-w)^{x_2}}{w^n} \frac{{\rm d}w}{w-z} \] where $\Gamma_1$ is a small counterclockwise path around 1 and $\Gamma_{0,z}$ is a counterclockwise path around 0 which contains $\Gamma_1$. For large parameter $L$, $n = \eta L$, $x = \nu L$, $t = \tau L$ (space and time are scaled the same way; hydrodynamic limit), we get \[ \approx \int\int \exp(L(F(w) - F(z))) \frac{{\rm d}z\, {\rm d}w}{w-z} \] where \[ F(w) = \tau w + \nu \ln(1-w) - \eta \ln w. \] We want to deform the contours so that ${\rm Re}\, F(w) < F_0$ and ${\rm Re}\, F(z) > F_0$ for some constant $F_0$. We have \[ {\rm Re}\, F(w) = \tau {\rm Re}\, w + \nu \ln|1-w| - \eta \ln|w|. \] Near 0, this function goes to $+\infty$, and near 1, it goes to $-\infty$. The function also goes to $\pm \infty$ for $w\to \pm\infty$. Also we have \[ F'(w) = \tau - \frac{\nu}{1-w} - \frac{\eta}{w} = \frac{\tau(1-w)w - \nu w - \eta(1-w)}{w(1-w)} \] So either the critical points of $w$ are both real-valued $r_0 \le r_1$ in between 0 and 1, or are both complex-valued in the region $0 < {\rm Re} < 1$. In the first case, shrink the $z$-contour so that ${\rm Re}\, F(z) = {\rm Re}\, F(r_1)$ and expand the $w$-contour so that ${\rm Re}\, F(w) = {\rm Re}\, F(r_0)$. The case when both critical values are not real-valued is more complicated. This corresponds to the condition \[ |\sqrt{\tau} - \sqrt{\eta}| < \sqrt{\nu} < \sqrt{\tau} + \sqrt{\eta}. \] Let $\Omega, \ol{\Omega}$ be these critical points. In this case, we take $F_0 = {\rm Re}\, F(\Omega)$. Then \[ K(x_1, x_2) = \frac{1}{2\pi i} \int_{\ol{\Omega}}^{\Omega} (1-w)^{x_2 - x_1 - 1} {\rm d} w = \frac{1}{2\pi i} \int_{1-\Omega}^{1 - \ol{\Omega}} \frac{{\rm d}u}{u^{x_1 - x_2 + 1}} \] Evaluating this at $x_1 = x_2$, we get $\frac{1}{2\pi i} (\ln (1-\ol{\Omega}) - \ln(1-\Omega))$. The distance from 0 to $\Omega$ is $\sqrt{\eta/\tau}$ and the distance from $\Omega$ to 1 is $\sqrt{\nu / \tau}$. Consider the triangle formed by $0, \Omega, 1$. Call the angles $\pi_\nu, \pi_\tau, \pi_\eta$ at $0, \Omega, 1$, respectively. So $\rho_1(x) = K(x,x) \approx \pi_\eta / \pi$. How to draw a 3d picture? This is a plot of the height function $h(t,n,x)$, which is the number of particles to the right of $x$ at level $n$. We want to compute the limit shape \[ \lim_{L \to \infty} L^{-1} h(t L, \eta L, \nu L) =: H(\tau, \eta, \nu). \] We know that \[ H(\tau, \eta, \nu) = \int_{\nu}^{+\infty} \text{density of particles} = \frac{1}{\pi} \int_\nu^{+\infty} \pi_\eta(\tau, \eta, \nu) {\rm d}\nu. \] So $\dfrac{\partial H}{\partial \nu} = -\dfrac{\pi_\eta}{\pi}$. Furthermore, $( \nu \dfrac{\partial}{\partial \nu} + \eta \dfrac{\partial}{\partial \eta} + \tau \dfrac{\partial}{\partial \tau}) H = H$ by homogeneity. Also, we have \[ F'(\Omega) = \tau - \frac{\nu}{1-\Omega} - \frac{\eta}{\Omega} = 0. \] Taking partial derivatives, we get the relations \begin{align*} F''(\Omega) \frac{\partial \Omega}{\partial \nu} - \frac{1}{1-\Omega} &= 0\\ F''(\Omega) \frac{\partial \Omega}{\partial \eta} - \frac{1}{\Omega} &= 0\\ F''(\Omega) \frac{\partial \Omega}{\partial \tau} + 1 &= 0 \end{align*} Hence we get $(1-\Omega)\Omega_\nu = \Omega \Omega_\eta = \Omega_\tau$. So from \[ H = \frac{1}{\pi} \int_{\nu}^{+\infty} \pi_\eta {\rm d}\nu = \frac{1}{\pi} \int_\nu^{+\infty} {\rm Im}(\ln(1-\Omega)) {\rm d}\nu, \] we get \begin{align*} \frac{\partial H}{\partial \tau} &= {\rm Im}\, \frac{1}{\pi} \int_\nu^{+\infty} \frac{-1}{1-\Omega} \frac{\partial \Omega}{\partial \tau} {\rm d} \nu\\ &= {\rm Im}\, \frac{1}{\pi} \int_\nu^{+\infty} -\Omega_\nu {\rm d}\nu = \frac{1}{\pi} {\rm Im}\, \Omega\\ \frac{\partial H}{\partial \eta} &={\rm Im}\, \frac{1}{\pi} \int_\nu^{+\infty} \frac{-1}{1-\Omega} \frac{\partial \Omega}{\partial \eta} {\rm d}\nu\\ &= {\rm Im}\, \frac{1}{\pi} \int_\nu^{+\infty} \frac{-\Omega_\nu}{\Omega} {\rm d}\nu\\ &= \frac{1}{\pi} {\rm Im}\, \ln(\Omega)|_{+\infty}^\nu = \frac{1}{\pi}(\pi - \pi_\nu). \end{align*} Finally, we conclude that \[ H = \frac{1}{\pi}(-\nu \pi_\eta + \eta(\pi - \pi_\nu) + {\rm Im}\, \Omega). \] Note that $\Omega$ is a complex coordinate on the limit shape, which maps $(\tau, \eta, \nu)$ to the upper-half plane, and the derivation above was made easier by using such a coordinate. Now consider the random object (with parameter $\Omega$) $H^\circ(\Omega) = h(L\tau, L\eta, L\nu) - LH(\tau, \eta, \nu)$ obtained by subtracting the law of large numbers. This is called the {\bf Gaussian free field}. Then \begin{compactitem} \item $H^\circ(\Omega)$ is asymptotically Gaussian $N(0, \ln L)$ \item \[ \lim_{L \to \infty} \bE (H^\circ(\Omega_1) \cdots H^\circ(\Omega_k)) = \sum G(\Omega_{i_1}, \Omega_{i_2}) \cdots G(\Omega_{i_{k-1}, i_k}) \] where the sum is over all perfect matchings of $\{1, \dots, k\}$ and \[ G(z,w) = \frac{1}{\pi} \ln\left|\frac{z-w}{z-\ol{w}}\right| \] is Green's function for the Laplacian on the upper-half plane with 0 boundary condition. \end{compactitem} \subsection*{Acknowledgements} \begin{compactitem} \item Alejandro Morales provided diagrams and typed the notes for September 29, 2011. \item Tom Roby provided notes for November 22, 2011. \item Darij Grinberg pointed out some corrections. \end{compactitem} \end{document}