On the derivation of the homogeneous kinetic wave equation for a nonlinear random matrix model

TL;DR

This paper derives the homogeneous kinetic wave equation for a nonlinear system influenced by a Hermitian random matrix, using advanced mathematical tools to analyze the weakly nonlinear, infinite volume limit.

Contribution

It introduces a rigorous derivation of the kinetic wave equation for a nonlinear random matrix model, employing Weingarten calculus for the analysis.

Findings

Leading order dynamics described by the kinetic wave equation

Use of Haar measure and Weingarten calculus for estimates

Results applicable to weakly nonlinear, infinite volume regimes

Abstract

We consider a nonlinear system of ODEs, where the underlying linear dynamics are determined by a Hermitian random matrix ensemble. We prove that the leading order dynamics in the weakly nonlinear, infinite volume limit are determined by a solution to the corresponding kinetic wave equation on a non-trivial timescale. Our proof relies on estimates for Haar-distributed unitary matrices obtained from Weingarten calculus, which may be of independent interest.