# Seminars & Events for Probability Seminar

##### Polynomials and (finite) free probability

Recent work of the speaker with Dan Spielman and Nikhil Srivastava introduced the ``method of interlacing polynomials'' (MOIP) for solving problems in combinatorial linear algebra. The goal of this talk is to provide insight into the inner workings of the MOIP by introducing a new theory that reveals an intimate connection between the use of polynomials in the manner of the MOIP and free probability, a theory developed by Dan Voiculescu as a tool in the study of von Neumann algebras. I will start with a brief introduction to free probability (for those, like me, who are not operator theorists). In particular, I will discuss the two basic operations in free probability theory (the free additive and free multiplicative convolutions), and how they relate to the asymptotic eigenvalue distributions of random matrices.

##### A reverse entropy power inequality for log-concave random vectors

We make two conjectures concerning reverse entropy power inequalities in the log-concave setting, discuss some examples and sketch the proof that the exponent of the entropy of one dimensional projections of a log-concave random vector defines a 1/5-seminorm.

Joint work with Keith Ball and Piotr Nayar.

##### Harmonicity and Invariance on the Slice

The subject of this talk is the slice of the discrete cube - i.e. the uniform distribution over all binary vector of a certain weight, or probabilistically the product measure on the cube conditioned on having a specific sum. I will review the L_2 theory of the slice as well as a the following new result: functions of low degree have similar distribution on the slice and the corresponding product measure on the cube. The proof relates harmonicity to decomposition of increasing path in terms of a Markov chain and a reverse Markov chain. Based on a joint work with Yuval Filmus (http://arxiv.org/abs/1507.02713).

##### Probabilistic interpretation of conservation laws and optimal transport in one dimension

We consider partial differential equations that describe the conservation of one or several quantities, possibly taking an additional dissipation mechanism into account, set on the real line. Such models are for instance relevant in gas dynamics or in the study of road traffic. When the initial data of these conservation laws are monotonic and bounded, a probabilistic theory can be developed by interpreting the solutions as cumulative distribution functions on the line. The study of the associated stochastic processes and their approximations by interacting particle systems provides a Lagrangian description of the solution, that will be used to derive existence results together with numerical schemes, as well as estimates of stability and convergence to equilibrium or traveling waves.

##### Random regular digraphs: singularity and spectrum

**Please note special day (Tuesday).** We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with iid bounded edge weights. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph has degree linear in the number of vertices. Towards establishing the same result for the unweighted adjacency matrix, we prove that it is invertible with high probability, even for sparse digraphs with degree growing only poly-logarithmically.

##### The maximum of the characteristic polynomial of random unitary matrices

A recent conjecture of Fyodorov, Hiary & Keating (FHK) states that the maxima of the characteristic polynomial of random unitary matrices behave like the maxima of a specific class of Gaussian fields, the so-called log-correlated Gaussian fields. These include important examples such as branching Brownian motion and the 2D Gaussian free field. In this talk, we will highlight the connections between the two problems. We will outline the proof of the conjecture for the leading order of the maximum. We will also discuss the connections with the FHK conjecture for the maximum of the Riemann zeta function on the critical line. This is based on joint works with D. Belius (NYU), P. Bourgade (NYU), and A. Harper (Cambridge).

##### Hyperplane Arrangements and Stopping Times

**Please note special day (Tuesday, November 10). ** Consider a real hyperplane arrangement and let C denote the collection of the occuring chambers. Bidigare, Hanlon and Rockmore introduced a Markov chain on C which is a generalization of some card shuffling models used in computer science, biology and card games: the famous Tsetlin library used in dynamic file maintenance and cache maintenance and the riffle shuffles are two important examples of hyperplane walks. I introduce a strong stationary argument for this Markov chain, which provides explicit bounds for the separation distance. I will try to explain both the geometric and the probabilistic techniques used in the problem.

##### Bigeodesics in first-passage percolation

In first-passage percolation, we place i.i.d. continuous weights at the edges of Z^2 and consider the weighted graph metric. A distance minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics. In the `90s, Licea-Newman showed that, under a curvature assumption on the ``asymptotic shape,'' there are no bigeodesics with one end directed in some deterministic subset D of [0,2\pi) with countable complement. I will discuss recent work with Jack Hanson in which we show that there are no bigeodesics with one end directed in any deterministic direction, assuming the shape boundary is differentiable.

##### Random matrices, differential operators and carousels

The Sine_\beta process is the bulk limit process of the Gaussian beta-ensembles. We show that this process can be obtained as the spectrum of a self-adjoint random differential operator. The result connects the Montgomery-Dyson conjecture about the Sine_2 process and the non-trivial zeros of the Riemann zeta function, the Hilbert-Polya conjecture, and de Brange’s approach of possibly proving the Riemann hypothesis. Our proof relies on the Brownian carousel representation of the Sine_beta process and a connection between hyperbolic carousels and first order differential operators acting on R^2 valued functions. [Joint with B. Virag (Toronto).]

##### Sudakov-type minoration

The classical Sudakov minoration principle gives a lower bound for suprema of Gaussian processes in terms of the metric entropy. We will discuss bounds of similar type for suprema of canonical processes and norms of logarithmically concave vectors. We will also present some applications based on chaining techniques.

##### Double dimers and tau-functions

The double dimer model is a variation of the classical dimer model consisting in superimposing two independent dimer configurations (perfect matchings) on a graph, thus creating an ensemble of non-intersecting loops. Kenyon has recently introduced and studied "anyonic" correlators for this model. We discuss the convergence (in the small mesh limit) of some of these correlators to the tau-functions appearing in the theory of isomonodromic (SU(2)) deformations.

##### Hypoelliptic Laplacian and Probability

The hypoelliptic Laplacian is a family of operators, indexed by b ∈ R_+, acting on the total space of the tangent bundle of a Riemannian manifold, that interpolates between the ordinary Laplacian as b → 0 and the generator of the geodesic flow as b → +∞. The probabilistic counterpart to the hypoelliptic Laplacian is a 1-parameter family of differential equations, known as geometric Langevin equations, that interpolates between Brownian motion and the geodesic flow. I will present some of the probabilistic ideas that explain some of its remarkable and often hidden properties. This will include the Ito formula for the corresponding hypoelliptic diffusion. I will also explain some of the applications of the hypoelliptic Laplacian that have been obtained so far.

##### First passage percolation on exponential of log-correlated Gaussian fields in two-dimensional lattice

I will discuss first passage percolation problems in two-dimensional lattice where the vertex weight is given by exponentiating log-correlated Gaussian fields. I will present some recent progress on the exponent for the weight of the geodesic including an upper bound as well as a non-universality result. This is based on joint works with Subhajit Goswami and Fuxi Zhang.

##### A universality result for the random matrix hard edge

The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and “quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by Jose Ramirez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with Patrick Waters.

##### The Markov sequence problem and the hypergroup property

The Markov sequence problem originates in work of Bochner concerning Markov processes on the sphere. The ultraspherical polynomials arise as eigenfunctions of certain Markov kernels. What sequences arise as the eigenvalue sequences of reversible Markov kernels having ultra spherical polynomials as their eigenfunctions? This question was answered by Bochner who described the extreme points of the convex set of such sequences. The problem has many connections with a wide range of problems in probability and analysis. This talk will present recent work growing out of a new approach to the Markov sequence problem by myself, Geronimo and Loss that originated in work on the Kac walk on high dimensional spheres.

##### Spectral statistics of random band matrices and quantum unique ergodicity

We prove the universality for the eigenvalue statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable to the linear size of the matrix. By relying on a mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.