TY - JOUR

T1 - Diffusions interacting through a random matrix

T2 - universality via stochastic Taylor expansion

AU - Dembo, Amir

AU - Gheissari, Reza

N1 - Funding Information:
The authors thank the anonymous referee for useful comments, and Ramon van Handel and Ofer Zeitouni for helpful conversations. This project was supported in part by NSF Grants #DMS-1613091, #DMS-1954337 (A.D.), and by the Miller institute for basic research in science (R.G.).
Publisher Copyright:
© 2021, The Author(s).

PY - 2021/8

Y1 - 2021/8

N2 - Consider (Xi(t)) solving a system of N stochastic differential equations interacting through a random matrix J= (Jij) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of (Xi(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.

AB - Consider (Xi(t)) solving a system of N stochastic differential equations interacting through a random matrix J= (Jij) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of (Xi(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.

KW - Gradient flows

KW - Langevin dynamics

KW - Markov semi-group

KW - Random matrices, Disordered systems

KW - Stochastic differential equations

KW - Universality

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U2 - 10.1007/s00440-021-01027-7

DO - 10.1007/s00440-021-01027-7

M3 - Article

AN - SCOPUS:85100455774

SN - 0178-8051

VL - 180

SP - 1057

EP - 1097

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -