Power law decay for systems of randomly coupled differential equations

TL;DR

This paper analyzes the long-term behavior of large systems of coupled differential equations with random coefficients, revealing a power-law decay in the solution norm under critical coupling conditions.

Contribution

It introduces a method to compute asymptotics of coupled differential systems with random matrices, highlighting a power-law decay at critical coupling.

Findings

Solution norm decays like t^{-1/2} at critical coupling

Provides formulas for normalized trace of functions of random matrices

Connects spectral properties of random matrices to differential equation dynamics

Abstract

We consider large random matrices $X$ with centered, independent entries but possibly different variances. We compute the normalized trace of $f(X) g(X^*)$ for $f,g$ functions analytic on the spectrum of $X$. We use these results to compute the long time asymptotics for systems of coupled differential equations with random coefficients. We show that when the coupling is critical the norm squared of the solution decays like $t^{-1/2}$.