The GPP seminar is part of the KUL--UCL Classical Analysis and Integrable Systems Seminar, co-organized at KULeuven and UCLouvain.
Information on the talks at KULeuven can be found here
Another adjoint seminar series is the monthly PIICQ Online Seminar on Integrable Probability, Classical and Quantum Integrability, see here
Organizers : Alexandre Lazarescu (UCLouvain)
UCL/KUL Integrable systems and Classical analysis Seminar (Sept. 2022– June 2023)
Tom Claeys (IRMP, UCL), 4 April (2023)
A biorthogonal ensemble characterizing the Log Gamma Polymer partition function
Abstract : The Log Gamma Polymer is a lattice path model introduced by Seppäläinen in 2012, in which the weight of each path is a product of independent inverse-Gamma distributed random variables. Depending on the choice of parameters, the partition function of the Log Gamma polymer has fluctuations that can be Gaussian or described by the Tracy-Widom distribution, the Baik-Ben Arous-Péché distribution, or the O'Connell-Yor polymer.
Relying on a Fredholm determinant identity of Borodin-Corwin-Remenik, I will show that the partition function is characterized by an intriguing biorthogonal ensemble involving Meijer G-functions. Our hope is that this biorthogonal ensemble will help to understand large deviations of the Log Gamma Polymer partition function. Based on joint work in progress with Mattia Cafasso.
Damir Ferizović (KUL), 14 March (2023)
Distributing points on a sphere, with a vengeance
Abstract : Imagine you place N electrons in a single shell around the nucleus: what would be the configuration of minimal potential energy for these points? This question and many more will not be answered in this talk – instead we will try to place points on the sphere, to make them as spread out as possible. We will give some small overview of why this is a useful thing to do, and then give a simple method to construct N points that are uniformly distributed, if N is a square number. Further, we will present a very recent result of the author on how to construct a sequence S=(z1,z2,…) such that any initial segment of S is uniformly distributed. Expect many pictures.
Giulio Ruzza, 1 March (2023)
Quantum KdV hierarchy, functions on partitions, and quasimodularity
Abstract : We consider the quantization of the KdV hierarchy in the first Poisson structure, as recently described in works by Buryak and Rossi.The spectral problem for the Hamiltonian operators is connected to interesting functions on partitions (shifted-symmetric functions,
appearing in representation theory and enumerative geometry) and to quasimodular forms.
Fudong Wang (University of South Florida), 13 December (2022)
An invitation to the spectral theory for fNLS soliton gases
Abstract : In this talk, we will discuss the spectral theory for the focusing nonlinear Schrodinger (fNLS) soliton gases. The analytic tool we use is the so-called finite-gap theory (an inverse scattering transform for quasi-periodic potential). Then by considering a thermodynamic type limit of the wavenumber-frequency relations for multi-phase (finite-gap) solutions of the focusing nonlinear Schrodinger equation, we will introduce the central object in the soliton gas theory: the nonlinear dispersion relations (NDR). The NDR will define the density of states which, in principle, contains all spectral and statistical properties of a soliton gas. Then, we will use the density of states to derive the average densities and fluxes for such fNLS soliton gases. After that, we will also discuss periodic gases and a realization problem (in the content of KdV). Finally, some open problems related to logarithmic potential theory will also be mentioned.
The talk is based on the joint work with Alexander Tovbis.
Guilherme Silva (Universidade de Sao Paolo, Brasil), 10 November (2022)
Integrablesystems for multipoint distributions of KPZ fixed points
Abstract : The KPZ universality conjecture is the belief that growth models in 1+1 dimensions exhibiting a 1: 2: 3 nontrivial scaling for its height, spatial location and temporal fluctuations should be described, in the large time limit, by an universal fluctuation field. The universal object itself is the KPZ fixed point, introduced for the first time by Matetski, Quastel and Remenik as a Markov process starting from a given initial condition.
In this talk, we discuss an alternative description of the KPZ fixed point and its periodic version in terms of nonlinear integrable systems such as the NLS and mKdV equations, generalizing the celebrated Tracy - Widom formula that relates the limiting eigenvalue distribution of Gaussian matrices with the second Painlevé equation.
The talk is based on joint work with Jinho Baik and Andrei Prokhorov.
Sofia Tarricone (UC Louvain), Tuesday October 18, 2022
Toeplitz determinants related to a discrete Painlevé II hierarchy
Abstract: The aim of this talk is to give a formula describing certain Toeplitz determinants, appearing in some random partitions models, in terms of solutions of a discrete Painlevé II hierarchy. This result is a nontrivial generalization of Borodin's formula for the discrete gap probabilities of the classical random partitions model when the measure is the Poissonized Plancharel measure. The result is obtained by using the Riemann-Hilbert problem (due to Baik-Deift-Johansson) associated to the family of orthogonal polynomials on the unit circle connected to the Toeplitz determinants of interest.
The talk is based on a joint work with Thomas Chouteau (Université d'Angers), soon on the ArXiv!
Giulio Ruzza (UCLouvain)
Cylindrical Toda equation for multiplicative statistics of the discrete Bessel process
Abstract: The discrete Bessel process is a discrete determinantal point process arising in connection with the distribution of the longest increasing subsequence of a random permutation. I will report on a joint work with Mattia Cafasso (arXiv:2207.01421) where we use a discrete Riemann-Hilbert approach to Fredholm determinants expressing multiplicative statistics of this process (following a general method due to Its-Izergin-Korepin-Slavnov in the continuous case and later adapted by Borodin-Deift to the discrete case). We show by this method that such expectations are tau functions for the cylindrical Toda equation. Further applications include a discrete version of the Amir-Corwin-Quastel integro-differential Painlevé II equation and asymptotic analysis.