Mixing of Glauber Dynamics on High Overlap Gibbs Measures
This provides the first rigorous guarantee of fast mixing for Glauber dynamics in the high-temperature SK model, a long-standing problem in statistical physics and computer science.
The authors prove that Glauber dynamics mixes in polynomial time for certain quadratic Gibbs measures with large external fields, including the Sherrington-Kirkpatrick model at any fixed finite inverse temperature, provided the external field strength exceeds a system-size-independent threshold.
We show fast mixing of Glauber dynamics for certain quadratic Gibbs measures with large external fields. The main ingredient is an overlap condition that allows us to control correlation matrices uniformly over all pinnings, by controlling norms of small submatrices of the interaction matrix. Using stochastic localization, we then obtain a lower bound on the spectral gap and, consequently, polynomial-time mixing of Glauber dynamics. As a direct application, we consider the Sherrington-Kirkpatrick model, whose interaction matrix is a scaled GOE matrix. For this model, we show that for any fixed finite inverse temperature $β$, there exists a strength of external field $θ$, not depending on the size of the system, for which Glauber dynamics mixes in polynomial time (with high probability on the draw of the interaction matrix).