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

Consider ( X i ( t ) ) solving a system of N stochastic differential equations interacting through a random matrix 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 μ independent of J , are universal, i.e., only depend on the choice of the distribution 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.