Random sampling for the monomer–dimer model on a lattice

Berg, J. van den; Brouwer, Rachel M.

doi:10.1063/1.533198

Cited by 15 publications

(7 citation statements)

References 12 publications

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“…For spin systems on lattice graphs, close connections between the mixing time of block dynamics and Glauber dynamics are known [42]. Such connections were for example applied to improve the mixing time of Glauber dynamics of the Monomer Dimer model on torus graphs [3]. Moreover, block dynamics were used to improve conditions for rapid mixing of Glauber dynamics on specific graph classes, such as proper colorings [17,19,20,46] or the hard-core model [19,46] in sparse random graphs.…”

Section: Block Dynamicsmentioning

confidence: 99%

A spectral independence view on hardspheres via block dynamics

Friedrich¹,

Göbel²,

Krejca³

et al. 2021

Preprint

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The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in dimensions. Up to a fugacity of < e/2 , the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure.Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime.

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Section: Block Dynamicsmentioning

confidence: 99%

A spectral independence view on hardspheres via block dynamics

Friedrich¹,

Göbel²,

Krejca³

et al. 2021

Preprint

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“…Note that even if the Markov random field is not monotone, our proof shows mixing time O(|V | log |V |) for censored single-site dynamics; this improves by a log factor Corollary 3.3 of Van den Berg and Brouwer [2].…”

Section: Resultsmentioning

confidence: 75%

Can Extra Updates Delay Mixing?

Peres

Winkler

2013

Commun. Math. Phys.

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We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in O(n log n) steps on an n-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of log n compared to updates at uniform random locations on an n-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations.

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“…[2,3]). This completes the definition of the couplings ν 1 (x 0 by composing these fixed-length couplings along the shortest path in the graph ( , S), as we did earlier in the case i = 1.…”

Section: Construction Of Full Couplingmentioning

confidence: 96%

“…[2] for a precise proof of the above elementary fact which we refer to as the composition of couplings.…”

Section: Preliminariesmentioning

confidence: 99%

Variable length path coupling

Hayes

Vigoda

2007

Random Struct Algorithms

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ABSTRACT:We present a new technique for constructing and analyzing couplings to bound the convergence rate of finite Markov chains. Our main theorem is a generalization of the path coupling theorem of Bubley and Dyer, allowing the defining partial couplings to have length determined by a random stopping time. Unlike the original path coupling theorem, our version can produce multistep (non-Markovian) couplings. Using our variable length path coupling theorem, we improve the upper bound on the mixing time of the Glauber dynamics for randomly sampling colorings.

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Random sampling for the monomer–dimer model on a lattice

Cited by 15 publications

References 12 publications

A spectral independence view on hardspheres via block dynamics

A spectral independence view on hardspheres via block dynamics

Can Extra Updates Delay Mixing?

Variable length path coupling

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