Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction

Chen, Zongchen; Liu, Kuikui; Vigoda, Eric

doi:10.48550/arxiv.2004.09083

Cited by 15 publications

(49 citation statements)

References 19 publications

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“…is the critical point for the uniqueness/non-uniqueness phase transition on the ∆-regular tree. By using Theorem 1.1 with known spectral independence calculations [CLV20b], together with a coupling argument of Hayes and Vigoda Moreover, there exists an algorithm which, given ε > 0, outputs (with constant probability) a (1 + ε)multiplicative approximation of Z G,λ in time Õδ (n 2 poly(1/ε)).…”

Section: Our Resultsmentioning

confidence: 99%

“…O(n log n/δ) from a warm-start. By [CLV20b,Theorem 7], the Gibbs distribution is (C, η)-spectrally independent with C = O(δ −1 ), η = λ 1 + λ .…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…• Many interesting spin systems, such as the hardcore model below critical fugacity ([ALOG20]), proper colorings of triangle free graphs of maximum degree ∆ with (1.763.. + δ)∆ colors ([FGYZ21, CGŠV21], and antiferromagnetic 2-spin systems on bounded degree graphs in the tree uniqueness regime (with some gap, [CLV20b,CLV20a]) are (C, η)-spectrally independent with C = O(1) and η ∈ [0, 1), possibly close to 1 inverse polynomially in q or the maximum degree ∆.…”

Section: Introductionmentioning

confidence: 99%

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Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

Jain,

Pham,

Vuong

2021

Preprint

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We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent q-spin system on a graph G = (V, E) with maximum degree ∆. Notably, for several interesting examples, our bound covers the entire regime of ∆ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments:• We show that for a triangle-free graph G = (V, E) with maximum degree ∆ ≥ 3, the Glauber dynamics for the uniform distribution on proper k-colorings with k ≥ (1.763 • • • + δ)∆ colors has spectral gap Ωδ (|V | −1 ). Previously, such a result was known either if the girth of G is at least 5 [Dyer et. al, FOCS 2004], or under restrictions on ∆ [Chen et. al, STOC 2021; Hayes-Vigoda, FOCS 2003].• We show that for a regular graph G = (V, E) with degree ∆ ≥ 3 and girth at least 6, and for any ε, δ > 0, the partition function of the hardcore model with fugacity λ ≤ (1 − δ)λ c (∆) may be approximated within a (1 + ε)-multiplicative factor in time Õδ (n 2 ε −2 ). Previously, such a result was known if the girth is at least 7 [Efthymiou et. al, SICOMP 2019].• We show for the binomial random graph G(n, d/n) with d = O(1), with high probability, an approximately uniformly random matching may be sampled in time O d (n 2+o(1) ). This improves the corresponding running time of Õd (n 3 ) due to [Jerrum-Sinclair, SICOMP 1989;Jerrum, 2003].

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Section: Our Resultsmentioning

confidence: 99%

“…O(n log n/δ) from a warm-start. By [CLV20b,Theorem 7], the Gibbs distribution is (C, η)-spectrally independent with C = O(δ −1 ), η = λ 1 + λ .…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

Jain,

Pham,

Vuong

2021

Preprint

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“…Many of the most successful algorithms have been based on Markov chain simulation. Particularly prominent is the familiar Glauber dynamics [Gla63] whose mixing time has remained an active area of research, with wonderful new results coming in recent years [CLV20a;CLV20b], prompted by pioneering work on high dimensional expanders [KO20;ALGV18]. Deterministic approaches have also been proposed, based on decay of correlations [Wei06] or Taylor expansion in a zero-free region of the partition function [PR17;BS17].…”

Section: Introductionmentioning

confidence: 99%

Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing

Konrad¹,

Jerrum²

2021

Preprint

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“…The work of [2] establishes that, for 2-spin systems, it suffices to bound the largest eigenvalue of the n × n influence matrix I where the (v, w) entry captures the influence of the fixed spin at vertex v on the marginal probability at vertex w; we explain this in more detail in Section 1.1. The running time of the result of [2] was further improved in [5], who also generalised the approach to antiferromagnetic 2-spin systems up to the tree-uniqueness threshold by showing how to utilize potential-function arguments that were previously used to establish SSM. Going beyond 2-spin systems, all of these methods become harder to control even well above the treeuniqueness threshold, q = ∆ + 1, which marks the onset of computational hardness (even for triangle-free graphs, see [8]).…”

Section: Introductionmentioning

confidence: 99%

Rapid Mixing for Colorings via Spectral Independence

Chen

Galanis

Štefankovič

et al. 2020

Preprint

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The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hardcore model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems.Let α * ≈ 1.763 denote the solution to exp(1/x) = x and let α > α * . We prove that, for any trianglefree graph G = (V, E) with maximum degree ∆, for all q ≥ α∆ + 1, the mixing time of the Glauber dynamics for q-colorings is polynomial in n = |V |, with the exponent of the polynomial independent of ∆ and q. In comparison, previous approximate counting results for colorings held for a similar range of q (asymptotically in ∆) but with larger girth requirement or with a running time where the polynomial exponent depended on ∆ and q (exponentially). One further feature of using the spectral independence approach to study colorings is that it avoids many of the technical complications in previous approaches caused by coupling arguments or by passing to the complex plane; the key improvement on the running time is based on relatively simple combinatorial arguments which are then translated into spectral bounds.

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Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction

Cited by 15 publications

References 19 publications

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing

Rapid Mixing for Colorings via Spectral Independence

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