Mathematics Home / 2016 Clifford Lectures

**This year's Clifford Lecturer: Pierre van Moerbeke (Universite catholique de Louvain and Brandeis University)**

Outline: During these last two decades, mathematicians and physicists have shown that seemingly unrelated questions, like eigenvalues of Random Matrices, shuffling questions of integers, tiling models in statistical physics, stochastic growth models, and other models all fluctuate in some appropriate limit according to a very small set of statistical distributions; the Tracy-Widom distribution and their variations appear as archtype examples of these universal distributions. They can be viewed as the « complex cousin » of the gaussian distribution which describes fluctuations of sums of independent random variables. Some of these models will be discussed in some detail, including the relation to integrable systems.

**LECTURE 1: Random Matrices, Permutations, Percolations, Tilings and all that…**

**LECTURE 2: Random Permutations, Tracy-Widom distributions and Integrable equations (KdV, Toda,…)**

**LECTURE 3: Asymptotic distributions for Domino Tilings of Aztec Diamonds**

**LECTURE 4: Lozenge Tilings of Hexagons**

**Invited speakers include:**

Estelle Basor (American Institute of Mathematics)

Ivan Corwin (Columbia University)

Benedek Valko (University of Wisconsin, Madison)

Paul Bourgade (Courant Institute, New York University)

Jonathan Novak (University of California, San Diego)

Brian Rider (Temple University)

Leonid Petrov (University of Virginia)

**Title: The Sine_beta Operator**

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.

**Title: Random operators at the edge(s)**

I’ll describe (random) Schrödinger and Diffusion type operators which govern the limit laws of the extremal points for various log-gas ensembles in dimension one. These ensembles extend well-known (and solvable) real/complex/quaternion matrix models to general “inverse-temperature”. As such the advertised limits extend the classical Tracy-Widom and related laws in a similar way.

**Title: Universality for random matrices beyond mean-field**

The goal of this talk is to explain new ideas to prove universality for a class of random band matrices. We will show that quantum unique ergodicity provides a key role in proving random matrix statistics for such non-mean field models.

**Title: Yang-Baxter integrability of random growth models**

I will describe a structure allowing to compute averages of observables of several 1d interacting particle systems from the Kardar--Parisi--Zhang universality class (such as the stochastic six vertex model, ASEP, various $q$-TASEPs, and random polymer models). This structure is based on the Yang-Baxter equation, and allows to study asymptotic fluctuations in inhomogeneous particle systems, in particular, in a TASEP-like model with slow and fast regions. This system displays universal Tracy-Widom fluctuations, but its limit shape exhibits new unusual phase transitions.

**Title: Integrable probability**

We will describe how certain integrable systems (Macdonald symmetric functions and quantum integrable systems) translate into probabilistic systems which are amenable to exact analysis. Through this route we discover universal phenomena which should hold true for much wider classes of probabilistic systems (e.g. the KPZ universality class).

**Title: Asymptotics of determinants of block Toeplitz matrices**

This talk will review what is known about the asymptotics of determinants of block Toepliz matrices. (These matrices have constant matrix blocks on the diagonals.) While theorems for these asymptotic expansions are similar to the scalar case, in the block case the constants in the expansions are at times difficult to compute in an explicit way. The talk will describe methods that make this problem more tractable and some recent applications of the results to dimer models.

**Title: The story of the Itzykson-Zuber integral**

The Harish-Chandra/Itzykson-Zuber integral is a ubiquitous special function which plays a key role in random matrix theory and related fields. I will explain and contextualize a conjecture of Itzykson and Zuber concerning the asymptotic behaviour of this function in a certain scaling limit, and describe the progress on this conjecture made by I. Goulden, M. Guay-Paquet, and myself. Our approach is a mixture of analysis and combinatorics, and makes surprising contact with Hurwitz theory, a classical branch of enumerative algebraic geometry. If time permits, I will explain the relationship between Schur polynomials and HCIZ integrals, and argue that this relationship can be used to apply HCIZ asymptotics to problems in asymptotic representation theory and integrable probability.