Schedule

• February 12

  Eugene Strahov (Hebrew University of Jerusalem)

  Determinantal processes related to products of random matrices

  Abstract: I will discuss a family of random processes in discrete time related to products of random matrices (product matrix processes). Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. I will explain that in certain cases product matrix processes are discrete-time determinantal point processes, whose correlation kernels can be expressed in terms of double contour integrals. This enables to investigate determinantal processes for products of random matrices in different asymptotic regimes, and to compute different probabilistic quantities of interest.

• February 19

  NO SEMINAR

• February 26

  No probability seminar. See Applied Math Colloquim for a related talk.

• March 5

  Leonid Petrov (University of Virginia)

  Nonequilibrium particle systems in inhomogeneous space

  Abstract: I will discuss stochastic interacting particle systems in the KPZ universality class evolving in one-dimensional inhomogeneous space. The inhomogeneity means that the speed of a particle depends on its location. I will focus on integrable examples of such systems, i.e., for which certain observables can be written in exact form suitable for asymptotic analysis. Examples include a continuous-space version of TASEP (totally asymmetric simple exclusion process), and the pushTASEP (=long-range TASEP). For integrable systems, density limit shapes can be described in an explicit way. We also obtain asymptotics of fluctuations, in particular, around slow bonds and infinite traffic jams caused by slowdowns.

• March 12

  Alexander Magazinov (Tel Aviv University)

  Percolation in the hard sphere model

  Abstract: In this talk I will focus on the hard sphere model in R^d, in which a random set of non-intersecting unit balls is sampled with an intensity parameter λ.

  Consider the graph in which the vertex set is the set of balls, and two balls are adjacent if they are at distance <= ε from each other. We will discuss the connectivity of this graph for large λ in dimensions d = 2 and 3. I will sketch the proof that the graph is highly connected when λ is greater than a certain threshold depending on ε. Namely, a cube annulus with inner radius L_1 and outer radius L_2 is crossed by this graph with probability at least 1 - C exp(-c L_1^{d - 1}). This answers (a variant of) a question by Bowen, Lyons, Radin and Winkler (2006) and strengthens a result by Aristoff (2014).

• March 19

  Benedek Valko (Wisconsin Madison)

  Random matrices, operators and analytic functions

  Abstract: The finite circular beta-ensembles and their point process scaling limit can be represented as the spectra of certain random differential operators. I will review these representations and present a couple of applications.

  Joint with B. Virag (Toronto/Budapest).

• April 2

  Samuel Watson (Brown)

  Relating a classical planar map embedding algorithm to Liouville quantum gravity and SLE(16)

  Abstract: In 1990, Walter Schnyder introduced a way to endow a simple planar triangulation with a wood --- a triple of spanning trees --- which gives rise to a combinatorially natural grid embedding of the triangulation. It turns out that a uniformly sampled wooded triangulation on n vertices converges in the large-n limit to a random fractal surface (called Liouville quantum gravity) together with a triple of intertwined fractal curves (called SLE(16)). We will motivate this result by describing Schnyder's algorithm and discussing some history of random planar map convergence results, and we will also explain the role of LQG and SLE in the story.

• April 9

  Tai Melcher (University of Virginia)

  Convergence rates for the empirical spectral distribution of Brownian motion on the unitary group

  Abstract: In 1997 Biane showed that Brownian motion on the unitary group U(N) converges as a process to the `free unitary Brownian motion' as N gets large. A corollary of this result is the convergence of the empirical spectral distribution of a unitary Brownian motion to a deterministic probability measure which can be described as the spectral measure of a free unitary Brownian motion. We will discuss recent results bounding the rates of convergence of these measures for large N and fixed time, and also as measure-valued paths.

  This is joint work with Elizabeth Meckes

• April 16

  NO SEMINAR

• April 23

  3:00 pm -- 4:00 pm, ROOM 2-361 (special time and place due to Simons Lectures)

  Scott Armstrong (NYU)

  Quantitative stochastic homogenization of linear elliptic PDE

  Abstract: I will discuss the large-scale asymptotics of solutions of linear elliptic equations with random coefficients. It is well-known that solutions converge (in the limit of scale separation) to those of a deterministic equation, a kind of law of large numbers result called "homogenization". In recent years obtaining quantitative information about this convergence has attracted a lot of attention. I will give an overview of one such approach to the topic based on variational methods, elliptic regularity, and ``renormalization-group'' arguments.

• April 30

  Joseph Najnudel (Cincinnati)

  Exponential moments of the argument of the Riemann zeta function on the critical line

  Abstract: We give, under the Riemann hypothesis, an upper bound for the exponential moments of the imaginary part of the logarithm of the Riemann zeta function on the critical line. Our result, which is related the fluctuations of the distribution of the zeros of zeta, is similar to a previous result by Soundararajan on the moments of the absolute value of zeta.

• May 7

  Yury Polyanskiy (MIT)

  Broadcasting on directed acyclic graphs

  Abstract: Consider an infinite directed acyclic graph (DAG) with a unique source node X. Let the collection of nodes at distance k from X be called the kth layer. At time zero, the source node is given a bit. At time k each node in the (k - 1)th layer inspects its inputs and sends a bit to its descendants in the kth layer. Each sent bit is flipped with a probability of error $\delta$. The goal is to be able to recover X with probability of error better than 1/2 from the values of all nodes at an arbitrarily deep layer k. The classical example of trees shows existence of a critical $\delta$ beyond which recovery is impossible. This talk is about locating this threshold for other graphs: random-like and regular 2D and 3D grids. A tacit conjecture stimulating this work is that broadcasting is impossible in 2D and possible in 3D grids. I will talk about our steps towards resolving it.

  Joint work with Anuran Makur and Elchanan Mossel.