Asymptotic behaviour of Painlevé transcendents and random matrix models

• Grant number: 21/10819-3
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: March 01, 2022
• End date: March 01, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Guilherme Lima Ferreira da Silva
• Grantee: Carla Mariana da Silva Pinheiro
• 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
• Associated scholarship(s): 23/14157-0 - Deformations of the Painlevé I hierarchy, BE.EP.DR

Abstract

Several local statistics of interacting particle systems can be predicted by the behaviour of eigenvalues of large random matrices. These eigenvalues form, in turn, a determinantal point process, and their statistical behaviour when the size of the matrix grows is governed by the asymptotic behaviour of their correlation kernel. On the other hand, such kernels are asymptotically described by solutions to Painlevé equations. The goal of this project is two-fold. First, we will study the Painlevé XXXIV equation, with aiming at finding asymptotics for their general solutions. Second, we will carry out the asymptotic analysis of the q-Laplace transforms of random matrix models, where the the Painlevé XXXIV transcendents will also play a major role. (AU)