Orthogonal polynomials and statistical mechanics through novel potential methods

Orthogonal polynomials (OPs) are classes of polynomials subject to certain orthogonality relations, often in L2 spaces on the line, in the plane or along the circle.

Just as Fourier series may be used to express periodic functions in a simple way, orthogonal polynomials are used to describe the solutions to a variety of problems in mathematics.I´m interested in uses of OPs in random matrix theory, specifically for an important class of random matrices whose eigenvalues form a two-dimensional Coulomb gas.

The theory of 1D Coulomb gases, corresponding to Hermitian random matrices, underwent a revolution in the 1990s, driven to a large extent by the introduction of the Riemann-Hilbert method for analyzing OPs.The Riemann-Hilbert method does not work in the planar setting, except in special cases when the OPs happen to satisfy additional (seemingly unrelated) orthogonality relations.

The goal of my project is to overcome this issue.

We recently introduced new methods which led to substantial progress on the 2D Coulomb gas model, but important mysteries remain.

For instance, our methods do not reveal the location of the zeros of the planar OPs, and physical predictions concerning the free energy in the Coulomb gas model remain mathematically unjustified.

In addition to these planar problems, I propose a curious method which uses Brownian motion to estimate the norm of OPs for irregular weights on the circle, which are beyond the reach of current methods.