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

Amir Dembo, Reza Gheissari*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1057-1097
Number of pages41
JournalProbability Theory and Related Fields
Issue number3-4
StatePublished - Aug 2021


  • Gradient flows
  • Langevin dynamics
  • Markov semi-group
  • Random matrices, Disordered systems
  • Stochastic differential equations
  • Universality

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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