## 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) |
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Pages (from-to) | 1057-1097 |

Number of pages | 41 |

Journal | Probability Theory and Related Fields |

Volume | 180 |

Issue number | 3-4 |

DOIs | |

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