The logarithmic sobolev inequality for discrete spin systems on a lattice

Stroock, Daniel W.; Zegarliński, Bogusław

doi:10.1007/bf02096629

Cited by 146 publications

(114 citation statements)

References 15 publications

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“…There are several possibilities. One approach is to use logarithmic Sobolev inequalities: the results on spatial dependencies in Subsection 2.3 imply a mixing condition which, in turn, following a quite general theory developed by Stroock and Zegarlinski (see [16]), could lead to a bound on the mixing time of order Volume × log(Volume). (We write could because there is an extra, quite subtle, condition which has to be checked to obtain such a bound from the Stroock-Zegarlinski theory: see Theorem 1 in the survey paper [7] by Frigessi, Martinelli and Stander).…”

Section: Description and Motivation Of The Methodsmentioning

confidence: 99%

Random sampling for the monomer–dimer model on a lattice

Berg

Brouwer

2000

Journal of Mathematical Physics

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In the monomer-dimer model on a graph, each matching (collection of non-overlapping edges) M has a probability proportional to λ |M| , where λ > 0 is the model parameter, and |M | denotes the number of edges in M . An approximate random sample from the monomer-dimer distribution can be obtained by running an appropriate Markov chain (each step of which involves an elementary local change in the configuration) sufficiently long. Jerrum and Sinclair have shown (roughly speaking) that for an arbitrary graph and fixed λ and ε (the maximal allowed variational distance from the desired distribution), O(|Λ| 2 |E|) steps suffice, where |E| is the number of edges and |Λ| the number of vertices of the graph. For sufficiently nice subgraphs (e.g. cubes) of the ddimensional cubic lattice we give an explicit recipe to generate approximate random samples in (asymptotically) siginificantly fewer steps, namely (for fixed λ and ε) O(|Λ| (ln |Λ|) 2 ).

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Section: Description and Motivation Of The Methodsmentioning

confidence: 99%

Random sampling for the monomer–dimer model on a lattice

Berg

Brouwer

2000

Journal of Mathematical Physics

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“…Stroock and Zegarliński [35,37,38] proved that the logartihmic-Sobolev constant is uniformly bounded provided given the Dobrushin-Shlosman mixing conditions (complete analyticity). Finally, Martinelli and Olivieri [30,31] obtained this for cubes under the more general condition of strong spatial mixing.…”

Section: Theorem 2 Let D ≥ 1 and Consider The Continuous-time Glaubementioning

confidence: 99%

Cutoff for the Ising model on the lattice

2012

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Abstract. Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L 1 on a system of size n is O(log n). periodic boundary conditions has cutoff at (d/2λ∞) log n, where λ∞ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating L 1 -mixing to L 2 -mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e.g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.

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“…The first of these measures the rate of the exponential decay as t → ∞ of the variance Var π (e tL f ) computed with respect to the invariant measure π, while the second measures instead the rate of decay of the relative entropy of e tL f w.r.t π (see, e.g., [1]). Advances in statistical physics over the past decade have led to remarkable connections between these two quantities and the occurence of a phase transition (see, e.g., [40,30,29,9,28,26]). As an example, on finite n-vertex squares with free boundary in the 2-dimensional lattice Z 2 , when h = 0 and β is smaller than the critical value β c , the spectral gap and the logarithmic Sobolev constant are Ω(1) (i.e.…”

Section: Introductionmentioning

confidence: 99%

Glauber Dynamics on Trees: Boundary Conditions and Mixing Time

Martinelli

Sinclair

Weitz

2004

Commun. Math. Phys.

124

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We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with (+)-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(log n) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where there are multiple phases but the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hardcore constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard-core lattice gas (independent sets). † An extended abstract of this paper appeared under the title "The Ising model on trees: Boundary conditions and mixing time" in

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The logarithmic sobolev inequality for discrete spin systems on a lattice

Cited by 146 publications

References 15 publications

Random sampling for the monomer–dimer model on a lattice

Random sampling for the monomer–dimer model on a lattice

Cutoff for the Ising model on the lattice

Glauber Dynamics on Trees: Boundary Conditions and Mixing Time

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