DeepAI

Optimal mixing for two-state anti-ferromagnetic spin systems

We prove an optimal Ω(n^-1) lower bound for modified log-Sobolev (MLS) constant of the Glauber dynamics for anti-ferromagnetic two-spin systems with n vertices in the tree uniqueness regime. Specifically, this optimal MLS bound holds for the following classes of two-spin systems in the tree uniqueness regime: ∙ all strictly anti-ferromagnetic two-spin systems (where both edge parameters β,γ<1), which cover the hardcore models and the anti-ferromagnetic Ising models; ∙ general anti-ferromagnetic two-spin systems on regular graphs. Consequently, an optimal O(nlog n) mixing time holds for these anti-ferromagnetic two-spin systems when the uniqueness condition is satisfied. These MLS and mixing time bounds hold for any bounded or unbounded maximum degree, and the constant factors in the bounds depend only on the gap to the uniqueness threshold. We prove this by showing a boosting theorem for MLS constant for distributions satisfying certain spectral independence and marginal stability properties.

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05/31/2021

Rapid mixing of Glauber dynamics via spectral independence for all degrees

We prove an optimal Ω(n^-1) lower bound on the spectral gap of Glauber d...
11/04/2021

Optimal Mixing Time for the Ising Model in the Uniqueness Regime

We prove an optimal O(n log n) mixing time of the Glauber dynamics for t...
03/29/2022

Uniqueness of the Gibbs measure for the anti-ferromagnetic Potts model on the infinite Δ-regular tree for large Δ

In this paper we prove that for any integer q≥ 5, the anti-ferromagnetic...
06/12/2018

Swendsen-Wang Dynamics for General Graphs in the Tree Uniqueness Region

The Swendsen-Wang dynamics is a popular algorithm for sampling from the ...
11/07/2021

Fast sampling via spectral independence beyond bounded-degree graphs

Spectral independence is a recently-developed framework for obtaining sh...
11/11/2020

Uniqueness of the Gibbs measure for the 4-state anti-ferromagnetic Potts model on the regular tree

We show that the 4-state anti-ferromagnetic Potts model with interaction...
11/07/2021

Simple Parallel Algorithms for Single-Site Dynamics

The single-site dynamics are a canonical class of Markov chains for samp...