A free boundary problem in potential theory and singularity distribution of solutions to Painlevé equations

• Grant number: 20/13183-0
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)
• Start date: November 01, 2020
• End date: October 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Guilherme Lima Ferreira da Silva
• Grantee: Victor Julio Alves de Souza
• 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): 22/12756-1 - Approximation theory on Riemann surfaces, BE.EP.DD

Abstract

The characterization of zeros of Laguerre polynomials in terms of minimal energy equilibrium problems is a quite classical result, dating back to the works of Stieltjes. In the mid-late 20th century, this type of characterization of zeroes of orthogonal polynomials on the positive real axis has been extended to a much larger class of orthogonal polynomials other than the Laguerre ones. However, considerably less is known about zero distribution of non-hermitian Laguerre-type orthogonal polynomials, which are orthogonal on appropriate contours of the plane, starting at the origin, instead of on the positive real axis. Despite that, recently such type of non-hermitian orthogonal polynomials was found to be of great relevance in the understanding of singularity location of several families of special functions that solve the Painlevé ordinary differential equations. This project aims at developing the potential theory on the plane that underlies the zero distribution and asymptotic behavior of such non-hermitian orthogonal polynomials, and then apply the constructed knowledge for the asymptotic understanding of solutions to Painlevé equations. (AU)