I will discuss certain determinants with respect to a sufficiently regular Jordan curve in the complex plane that generalize Toeplitz determinants which are obtained when the curve is the circle. This also corresponds to studying a planar Coulomb gas on the curve at inverse temperature beta =2. Under suitable assumptions on the curve we prove a strong Szego type asymptotic formula as the size of the determinant grows. The resulting formula involves the Grunsky operator built from the Grunsky coefficients of the exterior mapping function for the curve. As a consequence of our formula we obtain the asymptotics of the partition function on the curve. Interestingly, this formula involves the Fredholm determinant of the absolute value squared of the Grunsky operator which equals, up to a multiplicative constant, the so called Loewner energy of the curve.

Large deviations principles for the distribution of the empirical measure of the equilibrium measure for the Generalized Gibbs ensembles of the classical Toda chain introduced by H. Spohn. We deduce its almost sure convergence and characterize its limit in terms of the limiting measure of Beta-ensembles. Our results apply to general smooth potentials. This is joint work with Ronan Memin.

Article discussed: https://arxiv.org/abs/2103.04858

I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, and with sequential and parallel update, as well as some extensions to processes with memory length larger than 1. Based on joint work with Konstantin Matetski.

Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.

Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.

Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.

I will discuss how polynomials with a non-hermitian orthogonality on a contour in the complex plane arise in certain random tiling problems. In the case of periodic weightings the orthogonality is matrixvalued. In work with Maurice Duits (KTH Stockholm) the Riemann-Hilbert problem for matrix valued orthogonal polynomials was used to obtain asymptotics for domino tilings of the two-periodic Aztec diamond. This model is remarkable since it gives rise to a gaseous phase, in addition to the more common solid and liquid phases.

We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence and local universality of the covariance with droplet, flat and some random initial profiles. Furthermore, we show that also the geodesic tree is locally universal. These are joint works with Alessandra Occelli and Ofer Busani.

We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.

The role of (bi/multi/matrix) orthogonal polynomials in random matrices, integrable systems and combinatorics is well known. Our goal is to report on recent progress in the definition of suitable extensions of the notion of orthogonality where the polynomials are replaced by sections of appropriate line bundles on Riemann surfaces. We discuss their definition in the spirit of various generalizations of the Padé problem and the formulation of appropriate matrix Riemann Hilbert problems that allow to characterize them as well as control their asymptotic behaviour. Applications to Matrix Orthogonal Polynomials and the KP hierarchy will also be discussed.

I will introduce a family of marked determinantal point processes and associated conditional ensembles, in which information about a random part of a point configuration is encoded. Special cases of these conditional ensembles appear in the Its-Izergin-Korepin-Slavnov method and in the study of number rigidity. I will discuss general properties of these ensembles, and show how they lead to a strengthened notion of number rigidity for determinantal point processes induced by a certain class of orthogonal projections, including the sine, Airy, and Bessel point processes. The talk will be based on joint work with Gabriel Glesner.

We provide a detailed asymptotic description of this interaction, for the case of the modified KdV equation. Kinetic velocity equations are derived, and they will be shown to be ubiquitous in singular limits of a number of integrable systems, and in random matrix theory as well.

In this talk, I report on the asymptotics of the characteristic polynomials of the Gaussian β-ensembles for general β > 0. Based on the Dumitriu-Edelman matrix models for the Gaussian β-ensemble, I will present a probabilistic coupling between the characteristic polynomial, a Gaussian analytic function and a new object called the stochastic Airy function. This random entire function arise as the scaling limit of the characteristic polynomial at the spectral edge and its zero set is exactly the Airy-β point process. This is joint work with Elliot Paquette and our results are based on the study of the transfer matrix recurrence satisfied by the characteristic polynomials.