Asymptotic analysis of repulsive point processes and integrable equations

Introduction

The purpose of this project is to apply and develop robust mathematical methods for solving asymptotic problems on repulsive point processes and partial differential equations. The point processes considered will mostly be taken from the theory of random matrices, such as the eigenvalues of random normal matrices.

Point Processes

We will also consider discrete point processes with a more combinatorial structure, such as lozenge tilings of a hexagon. These models are used in various fields, including:

• Neural networks
• Multivariate statistics
• Nuclear physics
• Number theory

Therefore, they have been widely discussed in the physics and mathematics literature. We will investigate asymptotic properties of such processes as the number of points (or eigenvalues, or lozenges) gets large.

Common Features and Differences

All the point processes considered have in common an interesting feature: they are repulsive, in the sense that neighboring points repel each other. However, in other aspects, these processes are very different from one another, and some of them require completely novel techniques.

For example, the tiling models are related to non-Hermitian matrix-valued orthogonal polynomials, and two-dimensional point processes are out of reach of standard methods when the rotation-invariance is broken. An important part of the project is to develop novel techniques to analyze these point processes.

Integrable Partial Differential Equations

The last part of the project focuses on integrable partial differential equations. The objective is to develop a new approach for solving long-standing problems with time-periodic boundary conditions.

Methodology

One of the main tools we will use is the Deift-Zhou steepest descent method for Riemann-Hilbert (RH) problems. By solving new problems using this approach, this project will contribute to the development of the method itself.

Impact

Since the range of applicability of RH methods is very broad, these new techniques are likely to have an impact on a wide spectrum of scientific questions.