# Randomly coupled differential equations with elliptic correlations

## TL;DR

This paper studies the long-term behavior of large systems of linear differential equations with random coefficients, incorporating elliptic correlation structures, and provides explicit formulas for their asymptotic decay properties.

## Contribution

It generalizes previous models by including elliptic correlations among coefficients and derives precise asymptotic formulas for the trace of functions of the random matrix and its conjugate.

## Key findings

- Proved explicit decay formulas as a function of asymmetry.
- Extended analysis to correlated random matrices.
- Corrected previous formulas on time decay.

## Abstract

We consider the long time asymptotic behavior of a large system of $N$ linear differential equations with random coefficients. We allow for general elliptic correlation structures among the coefficients, thus we substantially generalize our previous work [14] that was restricted to the independent case. In particular, we analyze a recent model in the theory of neural networks [27] that specifically focused on the effect of the distributional asymmetry in the random connectivity matrix $X$. We rigorously prove and slightly correct the explicit formula from [28] on the time decay as a function of the asymmetry parameter. Our main tool is an asymptotically precise formula for the normalized trace of $f(X) g(X^*)$, in the large $N$ limit, where $f$ and $g$ are analytic functions.