Exponentially Slow Mixing of the Low Temperature SK Model

TL;DR

This paper proves that the low-temperature dynamics of the SK model have an exponential mixing time in system size, contrasting with physics predictions of a stretched exponential, and clarifies the conditions under which these predictions apply.

Contribution

It provides a short proof that the mixing time is exponential at low temperatures, based on recent results on gapped spin configurations, clarifying the applicability of physics predictions.

Findings

Mixing time is exponential in system size at low temperature.

Physics predictions of stretched exponential mixing time do not hold for worst-case initial conditions.

The result depends on recent findings about gapped spin configurations.

Abstract

We give a short proof that low-temperature dynamics for the Sherrington-Kirkpatrick model have mixing time exponential in the system size, based on the recently proved existence of gapped spin configurations by (Minzer-Sah-Sawhney 2023, Dandi-Gamarnik-Zdeborov\'a 2023). This result is in contrast with a well established physics prediction which posits a stretched exponential mixing time of order $e^{N^{1/3 \pm o(1)}}$. Our proof clarifies that this prediction cannot apply to mixing from worst case initial conditions, but should presumably be understood to concern dynamics from a suitably random initialization.