Representation and Poly-time Approximation for Pressure of $$\mathbb {Z}^2$$ Lattice Models in the Non-uniqueness Region

Adams, Stefan; Briceño, Raimundo; Marcus, Brian; Pavlov, Ronnie

doi:10.1007/s10955-015-1433-4

Cited by 25 publications

(8 citation statements)

References 44 publications

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“…Date: 6/29/2018. 1 results apply to a wide class of statistical mechanics models, including the ferromagnetic Potts and hard-core models. The following theorem is representative of our results.…”

Section: Introductionmentioning

confidence: 89%

“…where Z(G, z) is the polymer partition function defined in (1). Under the conditions for which we obtain an FPTAS for Z(G, z) we obtain an efficient sampling algorithm.…”

Section: 1mentioning

confidence: 99%

“…An ǫ-relative approximation to a complex number Z = 0 is a complex numberẐ = 0 so that e −ǫ ≤ Ẑ Z ≤ e ǫ and the angle between Z andẐ viewed as vectors in the complex plane is at most ǫ 1. The Ursell function is often defined over ordered multisets with a additional factor 1/k!.…”

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confidence: 99%

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Algorithmic Pirogov-Sinai theory

Helmuth

Perkins

Regts

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice Z d and on the torus (Z/nZ) d . Our approach is based on combining contour representations from Pirogov-Sinai theory with Barvinok's approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of Z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus (Z/nZ) d at sufficiently low temperature. 1 n log Z T d n (λ) of the hard-core model on Z d . That is, we would like an algorithm which for any ǫ > 0 outputs a number η ∈ [f d (λ) − ǫ, f d (λ) + ǫ] and whose running time grows as slowly as possible as a function of 1/ǫ. Gamarnik and Katz [28] gave such an algorithm running in time polynomial in 1/ǫ for λ small enough that the strong spatial mixing holds; this condition implies the hard-core model is in the uniqueness regime. Adams, Briceño, Marcus, and Pavlov [1] gave a polynomial-time algorithm for approximating the free energy of the hard-core (and several other) models on Z 2 in a subset of the uniqueness regime. More interestingly, their results also apply for the hard-core and Widom-Rowlinson models on Z 2 in a subset of the non-uniqueness regime. The latter result is of a similar spirit to the results of this paper.

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Section: Introductionmentioning

confidence: 89%

“…where Z(G, z) is the polymer partition function defined in (1). Under the conditions for which we obtain an FPTAS for Z(G, z) we obtain an efficient sampling algorithm.…”

Section: 1mentioning

confidence: 99%

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Algorithmic Pirogov-Sinai theory

Helmuth

Perkins

Regts

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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“…Analogues of the Kolmogorov-Sinai entropy in higher dimensions for Markov random fields, in which again conditioning on a measure in a restricted number of directions, or on lexicographic pasts, takes place, also have been considered, for example in [1,57]. Existence of such entropies turns out to be a weaker property than continuity.…”

Section: Entropiesmentioning

confidence: 99%

Markov and almost Markov properties in one, two or more directions

van Enter,

Ny,

Paccaut

2020

Preprint

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In this review-type paper written at the occasion of the Oberwolfach workshop Onesided vs. Two-sided stochastic processes (february 22-29, 2020), we discuss and compare Markov properties and generalisations thereof in more directions, as well as weaker forms of conditional dependence, again either in one or more directions. In particular, we discuss in both contexts various extensions of Markov chains and Markov fields and their properties, such as g-measures, Variable Length Markov Chains, Variable Neighbourhood Markov Fields, Variable Neighbourhood (Parsimonious) Random Fields, and Generalized Gibbs Measures.

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“…Proof. By adapting [1,Theorem 7.3] to the context of Gibbs (Z 2 , H, Φ)-specifications π, we know that if π is such that Q(π) < p c (Z 2 ), then π satisfies exponential SSM, where p c (Z 2 ) denotes the critical probability for Bernoulli site percolation on Z 2 and Q(π) is defined as Given q ∈ N, consider the graph H q as shown in Figure 11. The graph H q consists of a complete graph K q+1 and two other extra vertices a and b both adjacent to every vertex in the complete graph.…”

Section: Examplesmentioning

confidence: 99%

Strong Spatial Mixing in Homomorphism Spaces

Briceño

Pavlov

2017

SIAM J. Discrete Math.

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Given a countable graph G and a finite graph H, we consider Hom(G , H) the set of graph homomorphisms from G to H and we study Gibbs measures supported on Hom(G , H). We develop some sufficient and other necessary conditions on Hom(G , H) for the existence of Gibbs specifications satisfying strong spatial mixing (with exponential decay rate). We relate this with previous work of Brightwell and Winkler, who showed that a graph H has a combinatorial property called dismantlability if and only if for every G of bounded degree, there exists a Gibbs specification with unique Gibbs measure. We strengthen their result by showing that this unique Gibbs measure can be chosen to have weak spatial mixing, but we also show that there exist dismantlable graphs for which no Gibbs measure has strong spatial mixing.2010 Mathematics Subject Classification. 82B20, 68R10.

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Representation and Poly-time Approximation for Pressure of $$\mathbb {Z}^2$$ Lattice Models in the Non-uniqueness Region

Cited by 25 publications

References 44 publications

Algorithmic Pirogov-Sinai theory

Algorithmic Pirogov-Sinai theory

Markov and almost Markov properties in one, two or more directions

Strong Spatial Mixing in Homomorphism Spaces

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