Eikonal formulation of large dynamical random matrix models

TL;DR

This paper introduces an eikonal (Hamilton-Jacobi) formulation for large dynamical random matrix models, unifying various types including normal and non-normal dynamics, and enabling asymptotic calculations of integrals.

Contribution

It develops a novel Hamilton-Jacobi formalism for dynamical random matrices, extending the traditional eigenvalue trajectory approach with an optics-inspired analogy.

Findings

Unified description of diverse random matrix models.

Application to Brownian bridge dynamics for integral asymptotics.

Facilitates calculations of Harish-Chandra-Itzykson-Zuber integrals.

Abstract

Standard approach to dynamical random matrix models relies on the description of trajectories of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle(trajectories) and the Huygens principle (wavefronts), we formulate the Hamilton-Jacobi dynamics for large random matrix models. The resulting equations describe a broad class of random matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal dynamics. HJ formalism applied to Brownian bridge dynamics allows one for calculations of the asymptotics of the Harish-Chandra-Itzykson-Zuber integrals.