Dynamical phase transition for the mixing time of Langevin dynamics on spherical spin glasses

Determine the mixing-time behavior of Langevin dynamics, initialized from the uniform distribution on the sphere, for the mixed p-spin spherical spin glass with mixture function ξ, by proving that the mixing time is polynomial in N whenever ξ′(q) < q/(1−q) for all q in (0,1), and exponential otherwise (more precisely, when sup_{q∈(0,1)} (1−q)ξ′(q)/q > 1).

Background

The paper studies sampling from the Gibbs measure of mixed p-spin spherical spin glasses and reviews known and conjectured dynamical phenomena for associated Markov processes. Langevin dynamics is a canonical reversible diffusion whose mixing time is central to understanding algorithmic sampling.

Based on physics predictions derived from the Cugliandolo–Kurchan equations and thermodynamic heuristics, it is conjectured that the dynamics undergoes a sharp transition: fast (polynomial-time) mixing in a high-temperature regime and exponentially slow mixing in a low-temperature regime, delineated by inequalities involving the mixture function ξ. A full rigorous proof of this dichotomy remains open, though partial results establish shattering and some rapid mixing in limited regimes.