Schedule

• Sept 9

  Jacopo Borga (MIT)

  Long increasing subsequences in Brownian-type permutations

  Abstract: What is the behavior of the longest increasing subsequence in a uniformly random permutation? Its length is of order 2n^{1/2} plus Tracy--Widom fluctuations of order n^{1/6}. Its scaling limit is the directed geodesic of the directed landscape.
  
  This talk discusses how this behavior changes dramatically when one looks at universal Brownian-type permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants 1/2 < alpha< beta < 1 such that the length of the longest increasing subsequence in a random permutation of size n sampled from the Brownian separable permutons is between n^{alpha - o(1)} and n^{beta + o(1)} with high probability. We present numerical simulations which suggest that the lower bound is close to optimal and a very recent conjecture for the exact value of the exponent. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002).
  
  If time permits, we conclude by discussing some conjectures for permutations sampled from the skew Brownian permutons, a model of universal permutons generalizing the Brownian separable permutons: here, the longest increasing subsequences should be closely related with some models of random directed metrics on planar maps.
  
  Based on joint works with William Da Silva and Ewain Gwynne, and with Arka Adhikari, Thomas Budzinski, William Da Silva, and Delphin Sénizergues.

• Sept 16

  Ewain Gwynne (University of Chicago)

  Random conformal geometry in dimension $d\geq 3$

  Abstract: There has been enormous progress in the last few decades concerning random geometric objects in two dimensions which interact nicely with conformal maps. Such objects include Schramm-Loewner evolution (SLE), Liouville quantum gravity (LQG), and discrete analogs thereof. However, much less is known about analogs of these objects in dimension $d\geq 3$. I will give an overview of a few known results and many open problems concerning random geometry in dimension $d\geq 3$. Some of the known results come from recent joint works with Jian Ding and Zijie Zhuang, with Ahmed Bou-Rabee, and with Federico Bertacco. I will not assume any background knowledge about random geometry for $d=2$.

• Sept 23

  Sahar Diskin
  (Tel-Aviv University)

  Component sizes in percolation on finite regular graphs

  Abstract: A classical result by Erdős and Rényi from 1960 shows that the binomial random graph $G(n,p)$ undergoes a fundamental phase transition in its component structure when the expected average degree is around $1$ (i.e., around $p=1/n$). Specifically, for $p = (1-\epsilon)/n$, where $\epsilon > 0$ is a constant, all connected components are typically logarithmic in $n$, whereas for $p = (1+\epsilon)/n$ a unique giant component of linear order emerges, and all other components are of order at most logarithmic in $n$.
  
  A similar phenomenon — the typical emergence of a unique giant component surrounded by components of at most logarithmic order — has been observed in random subgraphs $G_p$ of specific host graphs $G$, such as the $d$-dimensional binary hypercube, random $d$-regular graphs, and pseudo-random $(n,d,\lambda)$-graphs, though with quite different proofs.
  
  This naturally leads to the question: What assumptions on a $d$-regular $n$-vertex graph $G$ suffice for its random subgraph to typically exhibit this phase transition around a critical probability $p=1/(d-1)$? Furthermore, is there a unified approach that encompasses these classical cases? In this talk, we demonstrate that it suffices for $G$ to have mild global edge expansion and (almost-optimal) expansion of sets of (poly-)logarithmic order in $n$. This result covers many previously considered concrete setups.
  
  We also discuss the tightness of our sufficient conditions.
  
  Joint work with Michael Krivelevich.

• Sept 30

  Allan Sly
  (Princeton, currently visiting MIT)

  Rotationally invariant first passage percolation: Concentration and scaling relations

  Abstract: For rotationally invariant first passage percolation on the plane, we use a multi-scale argument to prove stretched exponential concentration of the passage times at the scale of the standard deviation. Our results are proved for several standard rotationally invariant models of first passage percolation, e.g. Riemannian FPP, Voronoi FPP and the Howard-Newman model. As a consequence, we prove a version of the scaling relations between the passage times fluctuation and transversal fluctuations of geodesics. These are the first such unconditional results.

• Oct 7

  Nytia Mani
  (MIT)

  Fourth moment theorems for monochromatic subgraphs

  Abstract: Given a graph sequence $\{G_n\}_{n\ge1}$ and simple, connected subgraph $H$, denote by $T(H,G_n)$ the number of monochromatic copies of $H$ in a uniformly random vertex coloring of $G_n$ with $c \ge 2$ colors. For general $c$, we prove a central limit theorem for $T(H,G_n)$ with explicit error rates that arise from subgraph counts. Based on these counts, we show that for $c \ge 30$, convergence of the 4th moment is sufficient for $T(H, G_n)$ to enjoy a central limit theorem (and that it is always necessary).
  
  In the special case of 2 colors, we distil failures of the 4th moment phenomenon for $T(H, G_n)$ into easy-to-verify local properties of $\{G_n\}$. Along the way, we extend the fourth moment phenomenon to a broader class of Rademacher and Gaussian polynomials, which do not necessarily belong to a single Wiener chaos.
  
  Based on joint work with Sayan Das, Zoe Himwich, and Dan Mikulincer

• Oct 14

  Indigenous people's day

• Oct 21

  Konstantinos Kavvadias (MIT)

  Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion

  Abstract: The Liouville Brownian motion (LBM), introduced by Garban, Rhodes and Vargas, is a diffusion process evolving in a planar random geometry induced by the Liouville measure $\mu_h$ formally written as $\mu_h(dz) = \lim_{\epsilon \rightarrow 0} \epsilon^{\gamma^2/2} e^{\gamma h(z)} dz$, $\gamma \in (0,2)$, for a random field $h$ and it is the canonical diffusion process on a Liouville quantum gravity (LQG) surface. In this work, we establish upper and lower bounds for the heat kernel for LBM when $\gamma = \sqrt{8/3}$ in terms of the $\sqrt{8/3}$-LQG metric which are sharp up to a polylogarithmic factor in the exponential.

• Oct 28

  HT Yau (Harvard)

  Spectral statistics of Random regular graphs

  Abstract: In this lecture, we will review recent works regarding spectral statistics of the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices. Denote their eigenvalues by $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$ and let $\gamma_i$ be the classical location of the $i$-th eigenvalue under the Kesten-McKay law. Our main result asserts that for any $d \ge 3$ the optimal eigenvalue rigidity holds in the sense that \begin{align*} |\lambda_i-\gamma_i|\leq \frac{N^{o_N(1)}}{N^{2/3} (\min\{i,N-i+1\})^{1/3}},\quad \forall i\in \{2,3,\cdots,N\}. \end{align*} with probability $1-N^{-1+o_N(1)}$. In particular, the characteristic $N^{-2/3}$ fluctuations for Tracy-Widom law is established for the second largest eigenvalue. Furthermore, for $d \ge N^\varepsilon $ for any $\varepsilon > 0$ fixed, the extremal eigenvalues obey the Tracy-Widom law. This is a joint work with Jiaoyang Huang and Theo McKenzie.

• Nov 4

  Jasper Shogren-Knaak
  (NYU)

  A surface sum approach to lattice Yang—Mills in the large-N limit

  Abstract: Recently, Cao, Park, and Sheffield showed how to express Wilson loop expectations as sums of embedded maps for finite N-dimensional matrix group lattice Yang-Mills models. In this talk, I will present a sum of embedded maps formula for Wilson loop expectations in the large-N limit. In particular, to establish such a formula, I will explain how to interpret the master loop equation as a “peeling” exploration on maps and provide a description of “map cancellations”. Time permitting, I will illustrate how to use these tools to explicitly compute Wilson loop expectations in dimension two. This last result was previously established by Basu & Ganguly (2016) using different techniques.

  This is joint work with Jacopo Borga and Sky Cao.

• Nov 11

  Veteran's day

• Nov 18

  Jonathan Weitsman
  (Northeastern)

  Reflection Positivity and Chern-Simons Functional Integrals

  Abstract: We show that a mathematical version of the formal Chern-Simons functional integral of Witten for manifolds equipped with a reflection may be constructed in terms of a reflection positive functional, associated to the quadratic term in the Chern-Simons Lagrangian, on an algebra of functions on a Banach space of connections on the underlying 3-manifold. This construction yields a Hilbert space associated to a surface preserved by the reflection. A version of the cubic Bosonic interaction term in the Chern-Simons Lagrangian gives a self-adjoint operator on this Hilbert space, and by exponentiation, a unitary one parameter subgroup of operators. The vacuum expectation value of this one parameter subgroup is combined with an additional term associated to the ghost fields and their interaction, and an appropriate limit gives a partition function for the quantum field theory. This construction is nonperturbative. The theory is finite and does not require renormalization, as may be expected from perturbation theory. It is natural to ask whether the resulting partition function is related to the manifold invariants of Witten and Reshetikhin-Turaev, or more generally to topological quantum field theory.

• Nov 25

  Manan Bhatia
  (MIT)

  Bi-infinite geodesics in dynamical last passage percolation

  Abstract: Exponential last passage percolation (LPP) is a canonical planar directed model of random geometry in the KPZ universality class where the Euclidean metric is distorted by i.i.d. noise. One can also consider a dynamical version of LPP, where the noise is resampled at a constant rate, thereby gradually altering the underlying geometry. In fact, LPP is known to be noise sensitive in the sense that running the dynamics for a microscopic amount of time leads to a macroscopic change in the geometry. In this talk, we shall discuss the question of the existence of exceptional times in dynamical LPP at which bi-infinite geodesics exist. For static LPP, bi-infinite geodesics almost surely do not exist as was shown in Basu-Hoffman-Sly '18 and Balasz-Busani-Seppalainen '19.

  For dynamical LPP, we show that such exceptional times are at least very close to existing; namely, we give a subpolynomial lower bound ($(1/\log n)$) on the probability that there is an exceptional time $t\in[0,1]$ at which the origin lies on a geodesic of length $n$. In the other direction, for a dynamics on the related Brownian LPP model, we analyse `geodesic switches' to establish that the corresponding set of exceptional times almost surely has Hausdorff dimension at most 1/2-- we expect the correct dimension to be 0 as can be gathered by an intuitive non-rigorous argument.

• Dec 2

  William Da Silva (University of Vienna)

• Dec 9

  Mark Sellke (Harvard)