Let $P$ be a uniform random permutation matrix of size $N$ and let $\chi_N(z)= \det(zI - P)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically,
\[
\sup_{|z|=1}|\chi_N(z)|= N^{x_c + o(1)}
\]
with probability tending to one as $N\to \infty$, for a numerical constant $x_c\approx 0.652$. The main idea of the proof is to uncover an approximate branching structure in the distribution of (the logarithm of) $\chi_N$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. Unlike the well-studied \emph{CUE field} in which $P_N$ is replaced with a Haar unitary, the distribution of $\chi_N(z)$ is sensitive to Diophantine properties of the argument of $z$. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.

It was observed by Cordero-Erausquin, Fradelizi and Maurey that the classical Gaussian Poincare Inequality can be improved by a factor of 2 if we restrict our attention to even functions. They used this improved inequality to answer a geometric question asked by Banaszczyk about convexity of the Gaussian measure.

The Gaussian Poincare inequality is a special case of a more general variance inequality due to Brascamp and Lieb. In general, it is not clear how to improve this more general inequality in the even case. Again, this question turns out to be closely related to certain geometric problems. In this talk we will prove a sharp even Brascamp-Lieb inequality for measures of the form exp(-|x|^p). To do so we will also present a new weighted Poincare inequality which holds for ODD functions.

Based on joint work with Dario Cordero-Erausquin.

We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. Joint work with Aser Cortines (University of Zurich) and Oren Louidor (Technion).

I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.

Let $G=G(N,p)$ be an Erd\H{o}s--R\'enyi graph on $N$ vertices (where each pair is connected by an edge independently with probability $p$). We view $N$ as going to infinity, with $p$ possibly going to zero with $N$. What is the probability that $G$ contains twice as many triangles as we would expect? I will discuss recent progress on this ``infamous upper tail" problem, and more generally on tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of \emph{nonlinear large deviations}, and also connect with issues in extending the theory of \emph{graph limits} to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.

In R^n, the Bakry-Émery theorem states that the sharp constant in various functional inequalities for uniformly log-concave measures is no worse than the sharp constant for the Gaussian measure. As a consequence, uniformly log-concave measures inherit certain nice features of the Gaussian, like good measure concentration properties.

In this talk, I'll discuss quantitative stability estimates for the Bakry-Émery bound on logarithmic Sobolev and Poincaré constants. In particular, if a 1-uniformly log-concave measure has almost the same logarithmic Sobolev or Poincaré constant as the standard Gaussian measure, then it must (almost) split off a Gaussian factor. As a corollary, we obtain dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs combine Stein’s method, optimal transport, and simple variational arguments.

Joint work with Max Fathi.

Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a d-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. This is a joint work with E. Beckman, N. Frank, Y. Jiang, and M. Junge.

The majority vote process was one of the first interacting particle systems to be investigated and can be described as follows. There are two possible opinions at each site and that opinion switches randomly to the majority opinion of the neighboring sites. Also, at a different rate epsilon, the opinion at each site randomly changes due to noise.

Despite its simple dynamics, the majority vote process is difficult to analyze. In particular, on Z^d with d>1 and epsilon chosen small, it is not known whether there exists more than one equilibrium. This is surprising due to the close analogy between the majority vote process and the Ising model.

Here, we discuss work with Larry Gray on the majority vote process on the infinite tree with vertex degree d, where it is shown that, for small noise, there are uncountably many mutually singular equilibria, and that convergence to equilibrium occurs exponentially quickly from nearby initial states. Our methods rely on graphical constructions; they are quite flexible and can be used to obtain analogous results for other models, such as the stochastic Ising model on a tree.

This is the first part of a two-lecture presentation, and will concentrate on background and on existence of equilibria. Larry Gray will give the second part the following week, which will concentrate on convergence.