Universality and Random Matrix

The Harish-Chandra-Itzykson-Zuber integral is defined as the expectation of exp(Tr(AUBU*)) for A and B two self adjoint matrices and U Haar-distributed on the unitary/orthogonal/symplectic group. It was initially introduced by Harish-Chandra to study Lie groups. Since then, it has known many kinds of applications particularly in random matrix theory. In this talk we will explore the asymptotics of these integrals when one of the matrix remains of finite rank. We will see how to derive from these asymptotics large deviation principles for the largest eigenvalues for some random matrix models.

In this talk, I will revisit the problem of the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing a novel approach which combines field theoretical, probabilistic and integrable techniques. My goal will be to expand the program of the weak noise theory, which maps the large deviations onto a non-linear hydrodynamic problem, and to unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system.

This is based on the work arXiv:2103.17215 with Pierre Le Doussal

Local parametrix problems are ubiquitous in Riemann-Hilbert problems. In this talk I will present a technique that allows one to avoid them, provided certain a priori estiamtes are known. If time allows, some recent applications to Plancherel-Rotach asymptotics for orthogonal polynomials will be presented.

The Kac-Rice formula allows one to study the complexity of high-dimensional Gaussian random functions (meaning asymptotic counts of critical points) via the determinants of large random matrices. We present new results on determinant asymptotics for non-invariant random matrices, and use them to compute the (annealed) complexity for several types of landscapes. We focus especially on the elastic manifold, a classical disordered elastic system studied for example by Fisher (1986) in fixed dimension and by Mézard and Parisi (1992) in the high-dimensional limit. We confirm recent formulas of Fyodorov and Le Doussal (2020) on the model in the Mézard-Parisi setting, identifying the boundary between simple and glassy phases. Joint work with Gérard Ben Arous and Paul Bourgade.

Conformal blocks are fundamental inputs to the conformal bootstrap program of 2D conformal field theory and are closely related to four dimensional supersymmetric (SUSY) gauge theory via Alday-Gaitto-Tachikawa (AGT) correspondence. They also have intimate connections with isomonodromic tau functions.

In this talk, I will demonstrate a probabilistic construction of the 1-point torus conformal block and discuss how it transforms under the action of the modular group. This talk will be based on joint works with Guillaume Remy, Xin Sun and Yi Sun.

We consider the KPZ equation under the weak noise scaling. That is, we introduce a small parameter \epsilon

in front of the noise and let \epsilon \to 0. We prove that the one-point large deviation rate function has a 3/2 power law. Furthermore, by forcing the value of the KPZ equation at a point to be very large, we prove a limit shape of the KPZ equation as \epsilon \to 0. This confirms various predictions in the physics literature. Based on a recent joint work with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.

In this talk, I will first overview the spectral statistics of random regular graphs and random bipartite biregular graphs. Unlike standard Wigner matrices, the adjacency matrices of such random graphs have dependent entries, and the analysis of their spectra requires a good understanding of their combinatorial structures. We will focus on the global eigenvalue fluctuations of the two graph models and show that such fluctuations are determined by the cycle counts in the random graphs. Based on joint work with Ioana Dumitriu.

We present our result on stationary last passage percolation in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry. Joint work with D. Betea and P. Ferrari.

Structured moment determinants arise in several applications in random matrix theory and statistical mechanics. Aside from (pure) Toeplitz and Hankel determinants, there has been a growing interest in recent years in studying other (related) structured moment determinants, among which are Toeplitz+Hankel, bordered Toeplitz, and "2j-k"\"j-2k" determinants. In my talk, I will try to describe some aspects of these structured determinants as listed below.

• The orthogonality structures, recurrence relations, multiple integral formulae, and the Christoffel-Darboux identities for 2j-k and j-2k systems. These determinants are related to asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N), and O^-(2N). (joint work with N.Witte)

• The Riemann-Hilbert formulation for Toeplitz+Hankel determinants when a) Hankel symbol is supported on the unit circle, and b) Hankel symbol is supported on an interval on the real line. Some applications and the status of open problems will be discussed. (joint work with A.Its)

• Connection of bordered Toeplitz determinants with the solution of the Baik-Deift-Johansson Riemann-Hilbert problem for biorthogonal polynomials on the unit circle. The analogue of the strong Szego limit theorem for a class of bordered Toeplitz determinants, and the next-to-diagonal correlations of Ising model in the low-temperature regime. (joint work with A.Its, E.Basor, T.Ehrhardt, Y.Li)

In this talk I will introduce some Markov processes which are solvable by duality and how a duality relation can be constructed starting from Lie algebras. Moreover, we will see how the stationary measure of the processes in an open, out of equilibrium setting can be characterized.

Abstract: We will overview some limiting (zero-temperature) objects in the Kardar-Parisi-Zhang universality class. A subclass of models in this class is known as last passage percolation (LPP) and we will explain the limiting objects from the perspective of LPP. One of these objects is the parabolic Airy line ensemble, an infinite system of random continuous non-intersecting curves first constructed by Corwin-Hammond. The central probabilistic tool to analyse this object is its Brownian Gibbs property, an explicit description of conditional distributions in terms of Brownian bridges, which may be regarded as an integrable input that makes possible further non-integrable analysis. We will then discuss some results proved via Brownian Gibbs analysis in the last few years about the parabolic Airy line ensemble and related objects.

This talk will mostly be a general overview of connections between branching processes, random matrix polynomials, and certain number theoretic functions. The tools we use include integrable systems, Toeplitz/Hankel determinants, partition functions, symmetric function theory, Young diagrams, Gaussian multiplicative chaos.

Prominent work in the area includes papers of Keating, Fyodorov, Bouchaud, Arguin, Belius, Bourgade, Paquette, Zeitouni, Najnudel, Madaule, Chhaibi, Webb, Saksman, Nikula, Snaith, Harper, Radziwill, Sound., Katz, Sarnak, Remy, Rhodes, Vargas, Kahane, Bramson... (and many more, any omissions are down to my memory and not in any way a comment on the validity or importance of the work of others!)

We consider KPZ fixed point as the scaling limit of the continuous time TASEP model on the line with step initial condition. The resulting random field $h(t, y)$ called height function is commonly believed to be universal for a wide class of random growth processes and to be dependent only on initial conditions. We consider the formula for multitime distribution $\mathbb{P}(\cap_{k=1}^n h(t_k, y_k) < x_k) obtained as the result of series of works by Zhipeng Liu and Jinho Baik. It is given by the contour integral of Fredholm determinant of integrable integral operator. The one time distribution function $\mathbb{P}(h(t, y) < x)$ of the height function $h(t, y)$ is known to satisfy the KP equation in variables $t$, $y$ and $x$. We describe the differential equations appearing in the multitime case.

This is the joint work with Guilherme Silva and Jinho Baik.

In this talk I will present two different constructions of isomonodromic tau-functions. One as the determinant of a combination of Toeplitz operators, called a Widom constant; the other as a determinant of an integrable operator in the sense of IIKS. Painlevé equations will be my primary examples, although the techniques are applicable to any integrable system.