Deformations of the Painlevé I hierarchy

• Grant number: 23/14157-0
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate
• Start date: March 01, 2024
• End date: February 28, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Guilherme Lima Ferreira da Silva
• Grantee: Carla Mariana da Silva Pinheiro
• Supervisor: Mattia Cafasso
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Université d'Angers, France
• Associated to the scholarship: 21/10819-3 - Asymptotic behaviour of Painlevé transcendents and random matrix models, BP.DR

Abstract

Eigenvalues in unitary invariant random matrix ensembles are fascinating objects. Due to the symmetry of the model, the eigenvalues form a determinantal point process, and the most relevant statistics for them are encoded by a reproducing kernel in a very elegant way. However, the original kernel is not the only quantity of interest originated in such context. Applying a conditional thinning procedure to the set of eigenvalues, one obtains a deformed kernel and unlocks new possibilities of limit behaviors. The main goal of the present project, is to study a family of deformed kernels originated by the Painlevé I hierarchy kernel, and its multiplicative statistics.