Exponentially Slow Mixing of the Low Temperature SK Model
Mark Sellke
TL;DR
The paper proves that the low-temperature Glauber dynamics of the Sherrington–Kirkpatrick model mix exponentially slowly in system size, contradicting stretched-exponential predictions for certain initializations. It leverages recent existence results for gapped states to create bottlenecks in the state space and derives an energy-barrier lower bound around such a state. By combining a first-order energy gain with a controlled second-order term through a restricted-norm bound on the symmetric part of the disorder, it shows that the Gibbs measure assigns exponentially small mass to spheres around the gapped state, forcing an exponential mixing time via Cheeger’s inequality. The method extends to Wigner-type disorder under moment conditions via universality, reinforcing the computational hardness of low-temperature sampling.
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-Zdeborová 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.