Exponentially Slow Mixing of the Low Temperature SK Model

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.