2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) 2020
DOI: 10.1109/focs46700.2020.00124
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Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction

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Cited by 39 publications
(54 citation statements)
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“…Therefore, the Gibbs distribution π is the stationary distribution of the chain. Indeed, if Z′ is obtained from Z by deleting a vertex v then P(Z,Z)=1n(1+λ)andP(Z,Z)=λn(1+λ). Very recently, Chen et al [14] proved that the Glauber dynamics converges in time O(n2+32/δ) whenever λ(1δ)λc, where Δ is the maximum vertex degree and λc=(Δ1)Δ1/(Δ2)Δ is the tree uniqueness threshold. Now, by an easy calculation, λ<e/n implies that λ<λc.…”
Section: Markov Chainsmentioning
confidence: 99%
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“…Therefore, the Gibbs distribution π is the stationary distribution of the chain. Indeed, if Z′ is obtained from Z by deleting a vertex v then P(Z,Z)=1n(1+λ)andP(Z,Z)=λn(1+λ). Very recently, Chen et al [14] proved that the Glauber dynamics converges in time O(n2+32/δ) whenever λ(1δ)λc, where Δ is the maximum vertex degree and λc=(Δ1)Δ1/(Δ2)Δ is the tree uniqueness threshold. Now, by an easy calculation, λ<e/n implies that λ<λc.…”
Section: Markov Chainsmentioning
confidence: 99%
“…We would like to thank the anonymous referee for their helpful comments and for bringing [3, 14] to our attention. We also thank Ferenc Bencs for letting us know that the results of [8] lead to an FPTAS for the independence polynomial of bounded‐degree fork‐free graphs.…”
Section: Acknowledgmentsmentioning
confidence: 99%
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“…The phrase "rapidly mixing" indicates that the dynamics converges close to the Gibbs distribution in a number of updates that is polynomial in the size of the graph. In a recent paper, Chen, Liu and Vigoda [6] show that Glauber dynamics mixes rapidly up to a natural threshold for β, depending on the maximum degree of the graph, beyond which it was already known that the mixing time is exponential. Here we show that, by suitably restricting the underlying graph, we can have rapid mixing for all β.…”
Section: Introductionmentioning
confidence: 99%
“…Despite this clear complexity picture, prior to the introduction of spectral independence, the algorithms for λ < λ c (d) were based on elaborate enumeration techniques whose running times scale as n O(log d) [Wei06, LLY13, PR17, PR19]. The analysis of Glauber dynamics 1 using spectral independence in the regime λ < λ c (d) yielded initially n O(1) algorithms for any d [ALG20], and then O(n log n) for bounded-degree graphs [CLV21] (see also [CLV20]). More recently, Chen, Feng, Yin, and Zhang [CFYX21] obtained O(n 2 log n) results for arbitrary graphs G = (V, E) that apply when λ < λ c (∆ G − 1), where ∆ G is the maximum degree of G (see also [JPV21] for related results when ∆ G grows like log n).…”
Section: Introductionmentioning
confidence: 99%