Multiplicative statistics associated to higher-order discrete Bessel point processes

• Grant number: 25/06240-0
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: July 01, 2025
• End date: June 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Guilherme Lima Ferreira da Silva
• Grantee: Thomas Chouteau
• Supervisor: Alfredo Deaño Cabrera
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Universidad Carlos Iii De Madrid, Campus De Leganés, Spain
• Associated to the scholarship: 23/10533-8 - Deformations of orthogonal polynomials and integro-differential Painlevé equations, BP.PD

Abstract

This research project fits in a continuation of both the candidate's dissertation in integrable systems and of his research as a postdoc in collaboration with Professor G. Silva. It aims to study a new family of determinantal point processes and the integrable systems associated with them. Recent research on integrable systems and random matrices has extended classical results of these fields, connecting deformed models of determinantal point processes, random matrices and orthogonal polynomials with the integro-differential Painlevé II equation. The aim of this project is to introduce a new point process model built from generalizations of the Bessel function to extend the existing results for the discrete Bessel point process to a new class of point processes. One of the main objectives will be to relate these point processes with the theory of integrable equations by expressing their multiplicative statistics in terms of solutions of discrete Painlevé II and 2D-Toda equation's type. These new results will be based on the use of discrete Riemann-Hilbert problems associated with integrable operators, whose study will lead to the introduction of a new class of integrable systems. The study of asymptotics for this model will also be conducted in order to describe the large-scale behavior of the distribution of the largest particle in this deformed process. This project will take place at the Department of Mathematics at Universidad Carlos III de Madrid, and the candidate will join the GAMA research group (Grupo de Análisis Matemático Aplicado) under the supervision of Alfredo Deaño Cabrera. (AU)