Schedule

• February 11

  Marius Lemm (Harvard)

  On the averaged Green's function of an elliptic equation with random coefficients

  Abstract: We consider a divergence-form elliptic difference operator on the lattice Z^d, with a coefficient matrix that is a random perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to study the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which improves the key decay rate from -2d+ε to -3d+ε. (The optimal decay rate is conjectured to be -3d.) As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works. This is joint work with Jongchon Kim.

• February 18

  President's day

• February 25

  Gabor Lippner (Northeastern)

  Liouville's theorem via absolute monotonicity and random walks

  Abstract: Liouville's theorem, in its strong form, says that harmonic functions of polynomial growth are polynomials. I will describe a simple, new proof of this well-known theorem on Abelian and nilpotent Cayley graphs. The method relies on a surprising positivity phenomenon concerning squares of harmonic functions, that is an extension of the usual submartingale property. Joint work with Dan Mangoubi (Hebrew University).

• March 4

  Guillaume Dubach (Courant)

  Eigenvectors of non-Hermitian matrices

  Abstract: Eigenvectors of non-hermitian matrices are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables also quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and have been established for quaternionic Gaussian matrices.

• March 11

  Alex Hening (Tufts)

  Stochastic persistence and extinction

  Abstract: A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).

• March 18

  Jack Hanson (CUNY)

  Universality of the time constant for critical first-passage percolation on the triangular lattice

  Abstract: We consider first-passage percolation (FPP) on the triangular lattice with vertex weights whose common distribution function F satisfies F(0) = 1/2. This is known as the critical case of FPP because large (critical) zero-weight clusters allow travel between distant points in time which is sublinear in the distance. Denoting by T_n the first-passage time from 0 to the boundary of the box of sidelength n, we show existence of the time constant - the limit of T_n / log n - and find its exact value to be I / (2 (√ 3 π). (Here I = inf{x > 0 : F(x) > 1/2}.) This shows that the time constant is universal, in the sense that it is insensitive to most details of F. Furthermore, we find the exact value of the limiting normalized variance, which is also only a function of I, under the optimal moment condition on F.

• March 25

  Spring break

• April 1

  Guilherme Silva (Michigan)

  Connecting spectral curves for random matrix models to variational problems: an approach via quadratic differentials

  Abstract: A spectral curve for a matrix model is, in very loose terms, an equation with unknown being the Cauchy (a.k.a. Stieltjes) transform of the limiting spectral density. Sometimes also called master loop equation or string equation, it commonly appears as an algebraic equation, hence the name "curve" as it determines an algebraic curve. A classical situation is given by the celebrated semicircle law, whose Cauchy transform satisfies a very simple algebraic equation of degree 2.

  In this talk we plan to discuss spectral curves for the hermitian plus external source matrix model. Starting from the existence of the spectral curve, we construct a variational problem, involving a vector of three measures, that describes the underlying limiting spectral density. The first two measures satisfying this variational problem live on the real line and the third measure lives on a contour (a free boundary) whose determination is also part of the problem. Our key novel technique is to translate the determination of the solutions to the variational problem into the problem of geometrically describing trajectories of a canonical quadratic differential that lives on the underlying algebraic curve.

  As a consequence of our results, we are able to describe all possible critical local behaviors that can arise in the matrix model under consideration.

  This is a joint work with Andrei Martinez-Finkelshtein (Baylor University/Universidad de Almeria)

• April 8

  Usual time!

  Mykhaylo Shkolnikov (Princeton)

  The supercooled Stefan problem

  Abstract: We will consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. We will provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Holder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for the ordinary Stefan problem. In our second main theorem, we will establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present. Based on joint work with Francois Delarue and Sergey Nadtochiy.

• April 15

  Patriot's day

• Thursday, April 18

  NOTE: in 2-143 from 4:15-5:15!

  Rob Neel (Lehigh)

  Random walk approximations to sub-Riemannian diffusions, sub-Laplacians, and volumes

  Abstract: We study a variety of geometrically natural random walks on sub-Riemannian manifolds and their diffusion limits under parabolic scaling, which give, via their infinitesimal generators, second-order operators on the manifolds. On the probabilistic side, we give a general result on the convergence of random walks to a diffusion on a sub-Riemannian manifold, which is in the spirit of earlier work of Stroock-Varadhan on Euclidean space. Geometrically, one motivation is the lack of a canonical sub-Laplacian in sub-Riemannian geometry, and thus we are particularly interested in the relationship between the infinitesimal generators, the geodesic structure, and operators which can be obtained as divergences with respect to various choices of volume. In particular, we give a general criterion for the divergence of the gradient with respect to a smooth volume to also be realized as the infinitesimal generator coming from a horizontal random walk. We then determine how this turns out for some important classes of sub-Riemannian structures, such as the contact case (where things work out nicely) and Carnot groups (where one cannot expect uniqueness), as well as a "pathological" example of a 4-dimensional sub-Riemannian structure where there is a canonical choice of volume but the corresponding operator is not realized via a horizontal random walk.

  This work is joint with Ugo Boscain, Luca Rizzi, and Andrei Agrachev.

• April 22

  Oleg Zaboronski (Warwick)

  Pfaffian structures and Gap probabilities for reaction-diffusion systems

  Abstract: A class of reaction-diffusion systems in one dimension possesses Pfaffianor extended Pfaffian structure. This allows one to derive expressionsfor gap and non-crossing probabilities as Fredholm Pfaffians. Thesecan be analyzed using probabilistic methods pioneered by Marc Kac.In the talk I will review these results and discuss their applicationsto non-Hermitian random matrices, the Brownian web and the Net,symmetries and dualities for Markov processes.

• April 29

  3:00 pm in 2-147!!

  Tatyana Shcherbina (Princeton)

  Universality for random band matrices

  Abstract: Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.

• May 6

  Allan Sly (Princeton)

  The slow bond model with small perturbations

  Abstract: The slow bond model is the totally asymmetric simple exclusion process (TASEP) in which particles cross the edge at the origin at rate 1 - ε rather than at rate 1. Janowsky and Lebowitz asked if there was a global slowdown in the current for all ε>0. Using a range of theory and simulations two groups of physicists came to opposing conclusions on this question. With Basu and Sidoravicius this was settled, establishing that there is a slowdown for any positive ε. In the current work we illuminate the reason this problem was difficult to resolve using simulations, by analysing the effect of the perturbation as ε goes to 0 and showing it decays faster than any polynomial. This is joint work with Lingfu Zhang and Sourav Sarkar.

• May 13

  Andrew McIntyre (Bennington College)

  The determinant of the Laplacian on compact Riemann surfaces and holomorphic factorization

  Abstract: The regularized determinant of the Laplacian operator on a compact Riemann surface is a real-analytic function on any real-analytic family of such surfaces. If the family is a complex manifold, for example the Teichmuller moduli space of rigidified compact Riemann surfaces, we can ask about the relation to the complex structure. Were it not for the regularization, the determinant "ought" to be the modulus squared of a complex-analytic function on the moduli space, where this complex-analytic function is "morally" the determinant of the corresponding Dirac operator. Due to the regularization, this fails to be true by a factor of a "holomorphic anomaly", related to the classical value of the action for Liouville gravity. This line of reasoning was first pursued by Quillen and by Belavin-Knizhnik in the mid-1980s. The determinant of the Laplacian is expressed in terms of a Selberg zeta function; the "Dirac determinant" turns out to be related to a zeta function as well, but for an associated 3-manifold. The classical Liouville action turns out to be related to a regularized volume of the associated 3-manifold. In this talk, I will briefly outline more recent developments along these lines, and indicate some of the techniques of proof.