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

Amir Dembo; Reza Gheissari

doi:10.1007/s00440-021-01027-7

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

Amir Dembo, Reza Gheissari^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

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.

Original language	English (US)
Pages (from-to)	1057-1097
Number of pages	41
Journal	Probability Theory and Related Fields
Volume	180
Issue number	3-4
DOIs	https://doi.org/10.1007/s00440-021-01027-7
State	Published - Aug 2021

Funding

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.).

Keywords

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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10.1007/s00440-021-01027-7

Cite this

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

Abstract

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Keywords

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