Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 179 tok/s
Gemini 2.5 Pro 51 tok/s Pro
GPT-5 Medium 40 tok/s Pro
GPT-5 High 35 tok/s Pro
GPT-4o 103 tok/s Pro
Kimi K2 207 tok/s Pro
GPT OSS 120B 451 tok/s Pro
Claude Sonnet 4.5 35 tok/s Pro
2000 character limit reached

Spatial Mixing and Non-local Markov chains (1708.01513v1)

Published 3 Aug 2017 in cs.DM, math-ph, math.MP, and math.PR

Abstract: We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.

Citations (15)

Summary

We haven't generated a summary for this paper yet.

Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

Don't miss out on important new AI/ML research

See which papers are being discussed right now on X, Reddit, and more:

“Emergent Mind helps me see which AI papers have caught fire online.”

Philip

Philip

Creator, AI Explained on YouTube