Special Math/Physics Seminar

Random matrices, geometry and physics

Gaetan Borot

MPIM - Bonn

This lecture is a journey amidst large random matrices, 3d Chern-Simons theory and 2d quantum gravity.

Eigenvalues of large random matrices provide a computable playground to investigate universal laws occuring in the statistics of a large number of strongly coupled, repelling variables. Their universal is such that they can be observed in various situations in physics and mathematics. I will describe general results about asymptotic analysis of a class of matrix models called “beta-ensembles”, based on large deviation theory and functional analysis, and subsequent predictions about the Tracy-Widom law (i.e. the fluctuations of the maximum eigenvalue).

These general results can be applied to analyze the partition function of SU(N) Chern-Simons theory on simple 3-manifolds, namely quotients of S³ by a finite group of isometries. The large N limit is described by a spectral curve which we can compute, and identify with the spectral curve of a relativistic Toda chain of type ADE. From the physics perspective, we propose a generalization of the Gopakumar-Vafa conjecture, i.e. a correspondence between Chern-Simons theory on S³/Gamma, and topological strings on non-toric CY.

The all-order large N asymptotic expansion in these two problems are governed by a universal mathematical structure, called “topological recursion”. I will give other examples of its applications, related to volumes of moduli spaces and to conformal field theory.

If time allows, I will describe a recent result about 2d quantum gravity, which shows agreement of a multifractal spectral describing the nesting statistics of self-avoiding loop configurations in “large random discrete surfaces” vs. “Conformal Loop Ensembles coupled to Liouville quantum gravity”.