Asymptotic analysis of interacting particle systems and random matrix theory

• Grant number: 19/16062-1
• Support Opportunities: Research Grants - Young Investigators Grants
• Start date: March 01, 2020
• End date: February 28, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Guilherme Lima Ferreira da Silva
• Grantee: Guilherme Lima Ferreira da Silva
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated researchers: Daniel Ungaretti Borges ; Igor Mencattini ; Lun Zhang ; Maxym Yattselev ; Thaís Jordão
• Associated scholarship(s): 24/06997-1 - Asymptotics of Partition Functions of Coulomb Gases, BP.MS; 23/10533-8 - Deformations of orthogonal polynomials and integro-differential Painlevé equations, BP.PD; 23/06150-6 - Brownian Motion and Markov Chains, BP.IC; + associated scholarships 23/02674-0 - Random matrices and simulation of Coulomb gases, BP.IC; 23/01566-0 - Multiplicative Statistics for Orthogonal and Symplectic Ensembles, BP.MS; 21/10819-3 - Asymptotic behaviour of Painlevé transcendents and random matrix models, BP.DR; 21/09348-6 - Limiting theorems for eigenvalues of random matrices, BP.IC; 20/16783-8 - Random particle systems and Schur processes, BP.MS; 20/15699-3 - Asymptotic analysis for a model of products of coupled random matrices with arbitrary polynomial potential, BP.DD; 21/00358-9 - Asymptotic methods in mathematical Physics, BP.IC; 20/13183-0 - A free boundary problem in potential theory and singularity distribution of solutions to Painlevé equations, BP.DD; 20/02508-5 - Random Jacobi matrices, BP.MS; 20/02746-3 - Integrable systems and partition functions of random matrix models, BP.MS; 20/02506-2 - Asymptotic analysis of interacting particle systems and random matrix theory, BP.JP - associated scholarships

Abstract

In recent years random matrices have found applications in a vast number of different areas of science, such as telecommunications, high energy physics, number theory, machine learning, big data, dynamical systems, differential equations, computer science, among several others. The connection with interacting particle systems becomes natural when one asks questions about eigenvalues of large random matrices. Typically, such random eigenvalues interact with one another in a repulsive way, mimicking the interactions of several different and seemingly unrelated models from both equilibrium and non-equilibrium systems. Thus, one of the fundamental questions is to understand how the eigenvalues of a given large random matrix behave in different scaling regimes. This project proposes to explore several different facets of eigenvalues of large random matrices and other interacting particle systems. We will study how the thermodynamic limit of different interacting particle systems can be understood in terms of equilibrium problems on the plane, and use this information to carry out the asymptotic analysis of the models of interest, also expecting to unravel novel striking connections with integrable systems. (AU)

Scientific publications

(7)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

MARTINEZ-FINKELSHTEIN, ANDREI; SILVA, GUILHERME L. F.. Spectral Curves, Variational Problems and the Hermitian Matrix Model with External Source. Communications in Mathematical Physics, FEB 2021. (20/02506-2, 19/16062-1)

SILVA, GUILHERME L. F.; ZHANG, LUN. Large n Limit for the Product of Two Coupled Random Matrices. Communications in Mathematical Physics, v. 377, n. 3, JUN 2020. (19/16062-1, 20/02506-2)

GHOSAL, PROMIT; SILVA, GUILHERME L. F.. Universality for Multiplicative Statistics of Hermitian Random Matrices and the Integro-Differential Painleve II Equation. Communications in Mathematical Physics, v. 397, n. 3, p. 71-pg., 2022-11-10. (20/02506-2, 19/16062-1)

CELSUS, ANDREW F.; SILVA, GUILHERME L. F.. Supercritical regime for the Kissing polynomials (vol 255, 105408, 2020). Journal of Approximation Theory, v. 257, p. 3-pg., 2020-09-01. (19/16062-1, 20/02506-2)

MARTINEZ-FINKELSHTEIN, ANDREI; SILVA, GUILHERME L. F.. Spectral Curves, Variational Problems and the Hermitian Matrix Model with External Source. Communications in Mathematical Physics, v. 383, n. 3, p. 80-pg., 2021-02-09. (20/02506-2, 19/16062-1)

CELSUS, ANDREW F.; SILVA, GUILHERME L. F.. Supercritical regime for the kissing polynomials. Journal of Approximation Theory, v. 255, JUL 2020. (19/16062-1)

BAIK, JINHO; PROKHOROV, ANDREI; SILVA, GUILHERME L. F.. Differential Equations for the KPZ and Periodic KPZ Fixed Points. Communications in Mathematical Physics, v. 401, n. 2, p. 54-pg., 2023-04-13. (19/16062-1, 20/02506-2)