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

Abstract Consider $$(X_{i}(t))$$ (Xi(t)) solving a system of N stochastic differential equations interacting through random matrix $${\mathbf {J}} = (J_{ij})$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">J=(Jij) with independent (not necessarily identically distributed) coefficients. We show that the trajectories averaged observables $$(X_i(t))$$ , initialized from some $$\mu $$ xmlns:mml="http://www.w3.org/1998/Math/MathML">μ {J}}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">J are universal, i.e., only depend on choice distribution $$\mathbf {J}$$ its first and second moments (assuming e.g., sub-exponential tails). take general combinatorial approach to proving universality for dynamical systems coefficients, combining Taylor expansion moment matching-type argument. Concrete settings which our results imply include aging in spherical SK spin glass, Langevin dynamics gradient flows symmetric asymmetric Hopfield networks.