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arXiv · 2105.15005

Rapid mixing of Glauber dynamics via spectral independence for all degrees

Xiaoyu Chen·Weiming Feng·Yitong Yin·Xinyuan Zhang
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Abstract

We prove an optimal $Ω(n^{-1})$ lower bound on the spectral gap of Glauber dynamics for anti-ferromagnetic two-spin systems with $n$ vertices in the tree uniqueness regime. This spectral gap holds for all, including unbounded, maximum degree $Δ$. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack $δ\in(0,1)$: $\bullet$ $C(δ) n^2\log n$ mixing time for the hardcore model with fugacity $λ\le (1-δ)λ_c(Δ)= (1-δ)\frac{(Δ- 1)^{Δ- 1}}{(Δ- 2)^Δ}$; $\bullet$ $C(δ) n^2$ mixing time for the Ising model with edge activity $β\in\left[\frac{Δ-2+δ}{Δ-δ},\frac{Δ-δ}{Δ-2+δ}\right]$; where the maximum degree $Δ$ may depend on the number of vertices $n$, and $C(δ)$ depends only on $δ$. Our proof is built upon the recently developed connections between the Glauber dynamics for spin systems and the high-dimensional expander walks. In particular, we prove a stronger notion of spectral independence, called the complete spectral independence, and use a novel Markov chain called the field dynamics to connect this stronger spectral independence to the rapid mixing of Glauber dynamics for all degrees.

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Xiaoyu Chen, Weiming Feng, Yitong Yin, Xinyuan Zhang. 2021-11-19. Rapid mixing of Glauber dynamics via spectral independence for all degrees. https://arxiv.org/abs/2105.15005

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