Online Schedule

• September 14

  Russell Lyons (Indiana)

  Random Walks on Dyadic Lattice Graphs and Their Duals

  Abstract: Dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane. We use them to give examples of random rooted graphs that are stationary for simple random walk, but whose duals have only a singular stationary measure. This answers a question of Curien and shows behaviour different from the unimodular case. The consequence is that planar duality does not combine well with stationary random graphs. We also study harmonic measure on dyadic lattice graphs and show its singularity. Much interesting behaviour observed numerically remains to be explained. No background will be assumed for the talk. This is joint work with Graham White.

• September 21

  Eitan Bachmat (Ben-Gurion University)

  On maximal (weight) increasing subsequences

  Abstract: We will discuss the connection between the first order asymptotics of maximal weight increasing subsequences and comparison of natural (and implemented) airplane boarding policies.

  We then consider the behavior of weight fluctuations of maximal weight increasing subsequences by viewing them as discrete versions of maximal proper time curves in various space-time domains.

• September 28

  *** 1:15 - 2:15 pm (Special time)***

  Remi Rhodes (Marseille)

  Conformal Bootstrap in Liouville theory.

  Abstract: Liouville conformal field theory (denoted LCFT) is a 2-dimensional conformal field theory depending on a real-valued parameter γ and studied since the eighties in theoretical physics. In the case of the theory on the Riemann sphere, physicists proposed closed formulae for the n-point correlation functions using symmetries and representation theory, called the DOZZ formula (when n=3) and the conformal bootstrap (for n>3). A probabilistic construction of LCFT was recently proposed by David-Kupiainen-Rhodes-Vargas for γ in the half-open interval (0,2] and the last three authors later proved the DOZZ formula. In this talk I will present a proof of equivalence between the probabilistic and the bootstrap construction (proposed in physics) for the n point correlation functions with n greater or equal to 4, valid for γ in the open interval (0, √2). Our proof combines the analysis of a natural semi-group, tools from scattering theory and the use of Virasoro algebra in the context of the probabilistic approach (the so-called conformal Ward identities).

• October 5

  Jacapo Borga (Zurich)

  Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes

  Abstract: Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called tandem walks, are well-known to be related to each other through several bijections. In order to study their scaling and local limits, we introduce a further new family of discrete objects, called coalescent-walk processes and we relate them with the other previously mentioned families introducing some new bijections.

  We prove joint Benjamini-Schramm convergence (both in the annealed and quenched sense) for uniform objects in the four families. Furthermore, we explicitly construct a new random measure of the unit square, called the Baxter permuton, and we show that it is the scaling limit (in the permuton sense) of uniform Baxter permutations. We further relate the limiting objects of the four families to each other, both in the local and scaling limit case.

  To prove the scaling limit result, we show that the associated random coalescent-walk process converges in distribution to the coalescing flow of a perturbed version of the Tanaka stochastic differential equation. This result has connections with the results of Gwynne, Holden, Sun (2016) on scaling limits (in the Peanosphere topology) of plane bipolar triangulations.

  This is a joint work with Mickael Maazoun.

• October 12

  Indigenous People's Day (no seminar)

• October 19

  Jean-Christophe Mourrat (NYU)

  Mean-field spin glasses: beyond the replica trick?

  Abstract: Spin glasses are models of statistical mechanics encoding disordered interactions between many simple units. One of the fundamental quantities of interest is the free energy of the model, in the limit when the number of units tends to infinity. For a restricted class of models, this limit was predicted by Parisi, and later rigorously proved by Guerra and Talagrand. I will first show how to rephrase this result using an infinite-dimensional Hamilton-Jacobi equation. I will then present partial results suggesting that this new point of view may allow to understand limit free energies for a larger class of models, focusing in particular on the case in which the units are organized over two layers, and only interact across layers.

• October 26

  Alexander Kolesnikov (HSE)

  Blaschke-Santalo inequality for many functions and geodesic barycenters of measures

  Abstract: Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem.

  The talk is based on joint works with Elisabeth Werner.

• November 2

  Michael Damron (Georgia Tech)

  Critical first-passage percolation in two dimensions

  Abstract: In $2$d first-passage percolation (FPP), we place nonnegative i.i.d. weights $(t_e)$ on the edges of $\mathbb{Z}^2$ and study the induced weighted graph pseudometric $T = T(x,y)$. If we denote by $p = \mathbb{P}(t_e=0)$, then there is a transition in the large-scale behavior of the model as $p$ varies from $0$ to $1$. When $p < \frac{1}{2}$, $T(0,x)$ grows linearly in $x$, and when $p > \frac{1}{2}$, it is stochastically bounded. The critical case, where $p = \frac{1}{2}$, is more subtle, and the sublinear growth of $T(0,x)$ depends on the behavior of the distribution function of $t_e$ near zero. I will discuss my work over the past few years that (a) determines the exact rate of growth of $T(0,x)$, (b) determines the ``time constant'' for the site-FPP model on the triangular lattice and, more recently (c) studies the growth of $T(0,x)$ in a dynamical version of the model, where weights are resampled according to independent exponential clocks. These are joint works with J. Hanson, D. Harper, W.-K. Lam, P. Tang, and X. Wang.

• November 9

  Atilla Yilmaz (Temple)

  Stochastic homogenization of Hamilton-Jacobi equations in one dimension

  Abstract: After giving an introduction to the homogenization of Hamilton-Jacobi (HJ) equations, I will focus on HJ equations in one space dimension with Hamiltonians of the form $G(p) + \beta V(x,\omega)$, where $V$ is a stationary & ergodic potential of unit amplitude. The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive $G$. Under the extra condition that $G$ is a double-well function (for the sake of clarity and convenience), I will present a new and fully constructive proof of homogenization which yields a formula for the effective Hamiltonian $\overline H$ and clarifies the dependence of $\overline H$ on $G$, $\beta$ and the law of $V$.

• November 16

  Subhabrata Sen (Harvard)

  Large deviations for dense random graphs: beyond mean-field

  Abstract: In a seminal paper, Chatterjee and Varadhan derived an LDP for the dense Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained

  In this talk, we will explore large deviations for dense random graphs, beyond the ``mean-field" setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.

  Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.

• November 23

  Thanksgiving Break (no seminar)

• November 30

  *** 2:15 - 3:15 pm (Special time)***

  Dan Mikulincer (Weizmann Institute)

  Functional inequalities in Gauss space

  We will discuss how several known functional inequalities, such as log-Sobolev and Shannon-Stam, arise from general principles in stochastic analysis. This point of view will give rise to a unified framework from which one may study the stability of those inequalities. Several results in this direction will be presented with further applications to central limit theorems, normal approximations and optimal transport.

  Our method is based on an entropy-minimizing process from stochastic control theory, which allows us to express entropy as a solution to a variational problem.

  Based on joint works with Ronen Eldan, Alex Zhai and Yair Shenfeld

• December 7

  Benson Au (UCSD)

  Finite-rank perturbations of random band matrices via infinitesimal free probability

  Abstract: Free probability provides a unifying framework for studying random multi-matrix models in the large N limit. Typically, the purview of these techniques is limited to invariant or mean-field ensembles. Nevertheless, we show that random band matrices fit quite naturally in this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the 1/N correction. Our results allow us to extend previous work of Shlyakhtenko on finite-rank perturbations of Wigner matrices in the infinitesimal framework. For finite-rank perturbations of our model, we find outliers at the classical positions from the deformed Wigner ensemble.