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 -