Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

TL;DR

This paper demonstrates that the behavior of averaged observables in systems of stochastic differential equations with random matrix interactions is universal, depending only on the first two moments of the matrix distribution, using a stochastic Taylor expansion approach.

Contribution

The paper introduces a general combinatorial method combining stochastic Taylor expansion and moment matching to prove universality in dynamical systems with random coefficients.

Findings

Universality of averaged observables depends only on first two moments of the random matrix.

Applicable to models like spherical SK spin glass, Langevin dynamics, and Hopfield networks.

Method provides a unified approach to prove universality in various complex systems.

Abstract

Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $(X_i(t))$, initialized from some $\mu$ independent of $\mathbf J$, are universal, i.e., only depend on the choice of the distribution $\mathbf{J}$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.