Abstract: The last passage time field of exponential directed last passage percolations typically converges to the KPZ fixed point. In this talk, we consider the rare situation where the last passage time to a particular point is unusually large. We study the last passage time field under this unusual event and discuss how the limit differs from the KPZ fixed point. The analysis is based on multi-time joint distribution formulas. This talk is based on joint work with Dylan Cordaro and Tejaswi Tripathi.
Abstract: Stationary measures of a variety of growth models in the Kardar-Parisi-Zhang class can be described in terms of couples of interacting random walks or Brownian motions, called two-layer Gibbs measures. The Boltzman weights used to define them originate in branching rules satisfied by families of symmetric functions in the Macdonald hierarchy (Schur polynomials, Hall-Littlewood functions, (q)-Whittaker...). This framework applies to various models: interacting particle systems between boundary reservoirs, directed polymer models in a strip, last passage percolation, the stochastic six-vertex model. This talk will present an overview of the method, connecting with other approaches such as the Matrix Product Ansatz and Askey-Wilson processes. Time permitting, we will present recent developments related to random matrices.
Abstract: The purpose of this lecture will be to review some of the classical connections between random matrix theory and number theory, specifically the modelling of the value distribution of the Riemann zeta function near the critical line, and to explain some recent results concerning complex-valued joint moments of the Riemann zeta function and their evaluation in terms of Painlevé transcendents. The talk is intended for a broad maths audience, including graduate students, and does not require any prerequisites. Towards the end we will touch upon some ongoing joint work of the speaker with Fei Wei (Sussex).
Abstract: It is well-known that under a certain choice of weights random domino tilings of the Aztec diamond can be analyzed via Schur measures on partitions. In this talk we will discuss the model in which the parameters of the Schur measure ( or, equivalently, some weights of dominoes) are random themselves. We establish the limit shape and global fluctuations results via the technique of Schur generating functions.
Abstract: I will present a new method to characterize gap probabilities of discrete determinantal point processes in terms of Riemann-Hilbert problems. Simple examples of such discrete point processes arise in domino tilings of Aztec diamonds and lozenge tilings of hexagons. As a first illustration of our approach, we obtain a new explicit expression for the number of domino tilings of reduced Aztec diamonds in terms of Padé approximants, by solving the associated Riemann-Hilbert problem. As a second application, we obtain an explicit expression for the number of lozenge tilings of reduced hexagons in terms of Hermite-Padé approximants. This is based on joint work with Christophe Charlier.
Abstract: I will discuss a Markov chain on reverse plane partitions (of a given shape) which is closely related to the Toda lattice. This process has non-trivial Markovian projections and a unique entrance law starting from the reverse plane partition with all entries equal to plus infinity. I will also outline some connections with imaginary exponential functionals of Brownian motion, a random polymer model with purely imaginary disorder, interacting corner growth processes and discrete delta-Bose gas, and hitting probabilities for some low rank examples.
Abstract: We consider a N soliton solution of the focusing nonlinear Schrodinger equation. Generically solitons col- lide pairwise. We give conditions for the synchronous collision of N solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the sinc(x) function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. We derive a Central Limit Theorem for the fluctuations of the sinc(x) wave profile.
Abstract: Let's randomly choose a permutation of size 𝑁 uniformly and focus on observables such as the length of the longest increasing subsequence, the number of descents, the number of cycles of a given size, etc. The asymptotic behavior of these observables as 𝑁 becomes very large is well understood. In particular, it is easy to show that the joint distribution of traces of powers of permutation matrices is asymptotically of type Poisson. Now, if we consider not just a permutation, but a word formed by several independent uniform permutations, we know from the works of Nica, Puder, et al. that the asymptotic behavior of the trace of the word depends on the algebraic properties of the word considered. In this talk, I will review the uniform case and present a generalization to conjugacy invariant permutations. This talk is based on joint work with Mylène Maïda.
Abstract: The asymptotic behaviour of the partition function is one of the central questions of statistical mechanics. In our work we consider this problem when the external potential is complex valued and for a particular statistical-mechanical model, a Beta-ensemble. We prove a full 1/N expansion of the logarithm of the partition function, the so-called free energy. Our method can be regarded as an infinite dimensional version of the method of steepest descent for contour integrals. This is joint work with A. Guionnet and K. Kozlowski.
Abstract: Getting expansions for matrix integrals is an active topic within random matrix theory. When the coefficients of the expansions are related to geometrical or topological invariants, these expansions are called topological expansions. It is in general hard to show that topological expansions are not only formal power series but that they are convergent. In this talk, we will address this problem for a model of random unitary matrices that happens to be the partition function of the two-dimensional Yang-Mills theory with gauge group U(N). When the underlying surface is a torus, we have a full description of the topological expansion, show that it is related to the enumeration of ramified coverings of the torus and establish rigorously a string/gauge duality result predicted by Gross and Taylor in the nineties. This is joint work with Thibaut Lemoine (Collège de France).
Abstract: I will present a central limit theorem for the Lax matrix of some integrable systems which can be compared with the famous beta ensembles of random matrix theory, in the so-called high temperature setting. The key exemple of such a situation is the one of the Toda chain - a system of particles evolving through a nonlinear, nearest neighbors interaction ; which is linked with the high temperature beta ensemble on the real line. I will show that both types of models obey a CLT and that the limiting variances are linked through a simple relation. Based on the joint work with Guido Mazzuca https://arxiv.org/abs/2304.10323.
Abstract: I will first present a generic argument to derive large deviations of a stochastic process when large deviations of certain functionals of that process are available. I will then apply such a general argument to the analysis of the lower tail of the height functions of the stochastic six vertex model starting with step initial conditions. One of the main novelties will be a proof of weak logarithmic concavity of the cumulative distribution function of the height function. This is a joint work with Sayan Das and Yuchen Liao.
Abstract: In this talk we will revisit some properties of the so called two-point function for planar maps with bounded face degrees. In particular, we will see how it can be related to a specific family of orthogonal polynomials and how this allows us to give alternative analytical proofs of its integrable properties, as its determinantal representation and the integrable discrete equations that it satisfies, that were originally found by Bouttier-Di Francesco-Guitter via combinatorial methods. Based on work in progress with Jérémie Bouttier.
Abstract: We consider the dimer model, or equivalently domino tilings, on the Aztec diamond, and suppose that edges are assigned Fock’s weights. The main goal of this talk is to give a compact, explicit formula for the inverse Kasteleyn matrix, thus extending in this very general context previous results of the same kind; in particular, this gives an explicit expression for Boltzmann probabilities. Then, we will prove that the partition function admits a product form, and show how to recover Stanley’s celebrated formula as a specific case. Finally, we will show how our expression for the inverse Kasteleyn matrix allows to recover results about limit shapes. This is based on joint work with Cédric Boutillier.
Abstract: I will talk about a model of two dimensional random growth (namely, polynuclear growth) which can be translated into a probability law on integer partitions (by way of the RSK algorithm). We can find exact expressions for statistics of this model with algebraic tools, and compute fine asymptotics. I will focus on the model in half space with external sources driving growth at the edges. The limiting distribution of interface fluctuations in this model interpolates between different universal Tracy—Widom distributions from random matrix theory, and encodes solutions of the Painlevé II differential equation. At one point it matches a half-space version of the Baik—Rains distribution found by Barraquand, Krajenbrink and Le Doussal using methods from physics. Our approach uses connections between symmetric functions, matrix integrals, and Hankel determinants, plus a Riemann—Hilbert problem. Based on joint work with Mattia Cafasso, Alessandra Occelli and Daniel Ofner.