Isotropic Brownian motions over complex fields as a solvable model for May-Wigner stability analysis

TL;DR

This paper provides an exact solution for isotropic Brownian motions over complex fields, revealing their relation to May-Wigner stability analysis and deriving the full distribution of Lyapunov exponents.

Contribution

It introduces an exact solvable model for isotropic Brownian motions over complex fields and connects it to stability analysis and random matrix theory.

Findings

Exact joint distribution of finite-time Lyapunov exponents

Stability phase diagram for May-Wigner-like models

Connection to determinantal point processes and Hermitian matrix models

Abstract

We consider matrix-valued stochastic processes known as isotropic Brownian motions, and show that these can be solved exactly over complex fields. While these processes appear in a variety of questions in mathematical physics, our main motivation is their relation to a May-Wigner-like stability analysis, for which we obtain a stability phase diagram. The exact results establish the full joint probability distribution of the finite-time Lyapunov exponents, and may be used as a starting point for a more detailed analysis of the stability-instability phase transition. Our derivations rest on an explicit formulation of a Fokker-Planck equation for the Lyapunov exponents. This formulation happens to coincide with an exactly solvable class of models of the Calgero-Sutherland type, originally encountered for a model of phase-coherent transport. The exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices, and is also closely related to Hermitian matrix models with an external source.