# Dynamics for spherical spin glasses: disorder dependent initial conditions

## TL;DR

This paper analyzes the long-term behavior of spherical spin glass models under Langevin dynamics, linking the evolution of correlation functions to the geometric structure of Gibbs measures at different temperatures.

## Contribution

It derives the thermodynamic limit of correlation and response functions for spherical spin glasses with disorder-dependent initial conditions and relates their asymptotics to Gibbs measure geometry.

## Key findings

- Thermodynamic limit of correlation functions established
- Relation between dynamics and Gibbs measure structure shown
- FDT solution derived at high temperature

## Abstract

We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed $p$-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure $p$-spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature).