Locality of connective constants

Grimmett, Geoffrey; Li, Zhongyang

doi:10.48550/arxiv.1412.0150

Cited by 3 publications

(3 citation statements)

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“…Typically, however, the natural setting for such locality statements is not the class of transitive graphs, but the class of unimodular random graphs. Indeed, there are several interesting probabilistic quantities, most often related in some way to random walks, which have turned out to possess locality, mostly in the generality of unimodular random graphs: see [9,23,25,10,6,17] for specific examples, and [30,Chapter 14] for a partial overview. Therefore, it is natural to investigate Schramm's conjecture in the setup of unimodular random graphs and see what the proper notion of critical probability may be from the point of view of locality.…”

Section: Motivation and Resultsmentioning

confidence: 99%

On percolation critical probabilities and unimodular random graphs

Beringer¹,

Pete²,

Timár³

2017

Electron. J. Probab.

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Section: Motivation and Resultsmentioning

confidence: 99%

On percolation critical probabilities and unimodular random graphs

Beringer¹,

Pete²,

Timár³

2017

Electron. J. Probab.

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“…The results are modest but we believe that they open some interesting questions. Even though we will not discuss it here, let us mention that in recent years, Schramm's conjecture was also stated for self-avoiding walks in [Ben13], and that some partial results were obtained in [GL14,GL15].…”

Section: Motivationmentioning

confidence: 99%

A Note on Schramm’s Locality Conjecture for Random-Cluster Models

Duminil-Copin

Tassion

2019

Sojourns in Probability Theory and Statistical Physics - II

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In this note, we discuss a generalization of Schramm's locality conjecture to the case of random-cluster models. We give some partial (modest) answers, and present several related open questions. Our main result is to show that the critical inverse temperature of the Potts model on Z r × (Z 2nZ) d−r (with r ≥ 3) converges to the critical inverse temperature of the model on Z d as n tends to infinity. Our proof relies on the infrared bound and, contrary to the corresponding statement for Bernoulli percolation, does not involve renormalization arguments.

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“…Recently, the study of SAW on non-Euclidean lattices has received increasing attention. In a series of papers [15][16][17][18][19][20][21][22], Grimmett and Li initiated a systematic study of SAWs on transitive graphs. Their work is primarily concerned with properties of the connective constant.…”

Section: Introductionmentioning

confidence: 99%

Self-avoiding walks and polygons on hyperbolic graphs

Panagiotis¹

2019

Preprint

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We prove that for the d-regular tessellations of the hyperbolic plane by k-gons, there are exponentially more self-avoiding walks of length n than there are self-avoiding polygons of length n, and we deduce that the self-avoiding walk is ballistic. The latter implication is proved to hold for arbitrary transitive graphs. Moreover, for every fixed k, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion d − 1 − O(1/d) as d → ∞; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length n is comparable to the nth power of their connective constant. Some of these results were previously obtained by Madras and Wu [44] for all but finitely many regular tessellations of the hyperbolic plane.

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Locality of connective constants

Cited by 3 publications

References 18 publications

On percolation critical probabilities and unimodular random graphs

On percolation critical probabilities and unimodular random graphs

A Note on Schramm’s Locality Conjecture for Random-Cluster Models

Self-avoiding walks and polygons on hyperbolic graphs

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