Deformations of orthogonal polynomials and integro-differential Painlevé equations

• Grant number: 23/10533-8
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: March 01, 2024
• End date: February 28, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Guilherme Lima Ferreira da Silva
• Grantee: Thomas Chouteau
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/16062-1 - Asymptotic analysis of interacting particle systems and random matrix theory, AP.JP

Abstract

Recent findings on integrable systems and random matrices have extended classical results of these theories and linked deformed models of determinantal point processes, random matrices and orthogonal polynomials with integro-differential versions of Painlevé equations. Via the study of a deformed version of orthogonal polynomials for the cubic potential, we expect to introduce a novel integro-differential version for the Painlevé I equation. More precisely, this project is based on the Riemann-Hilbert approach for orthogonal polynomials and is related to a model introduced by Bleher and Deaño and extended works with Bahroumi and Yattselev. Studying these orthogonal polynomial ensemble and its partition function, the authors showed a connection between the free energy of this ensemble and the Painlevé I equation. Deforming this orthogonal polynomials model and studying the asymptotics of the deformed Riemann-Hilbert problem associated via the Deift-Zhou non linear steepest descent method, we expect to obtain an integro-differential version for the Painlevé I equation.