On the probability that self-avoiding walk ends at a given point

Duminil-Copin, Hugo; Glazman, Alexander; Hammond, Alan; Manolescu, Ioan

doi:10.1214/14-aop993

Cited by 12 publications

(30 citation statements)

References 22 publications

(32 reference statements)

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“…In [4], an upper bound on the closing probability of n −1/4+o (1) was proved in general dimension. Without significant modifications, the method used cannot prove an upper bound on this quantity that decays more rapidly than n −1/2+o (1) .…”

Section: Introductionmentioning

confidence: 99%

“…The next result, which is this conclusion, is the principal result of the present article. In order to prove Theorem 1.2, we will rework the method of [4], taking the opportunity to present this method in a general guise, with a view to future applications. Indeed, such an application has already been made, as we explain shortly.…”

Section: Introductionmentioning

confidence: 99%

“…It is the introduction and exposition of this general technique which is perhaps the principal advance of the present article. The use of the technique to prove Theorem 1.2 and its relation to the approach of [4] is remarked on at the end of Section 4.1.…”

Section: Introductionmentioning

confidence: 99%

“…In order to apply the snake method to reach an upper bound on the closing probability, such as Theorem 1.2, it is necessary to verify the sufficient condition in the method. This step is undertaken in this article, as it was in [4], by a technique of Gaussian pattern fluctuation.…”

Section: Introductionmentioning

confidence: 99%

“…We show that the closing probability Wn ||Γn|| = 1 that Γ's endpoint neighbours the origin is at most n −1/2+o(1) in any dimension d ≥ 2. The method of proof is a reworking of that in [4], which found a closing probability upper bound of n −1/4+o(1) . A key element of the proof is made explicit and called the snake method.…”

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confidence: 99%

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On self-avoiding polygons and walks: The snake method via pattern fluctuation

Hammond

2019

Trans. Amer. Math. Soc.

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For d ≥ 2 and n ∈ N, let Wn denote the uniform law on selfavoiding walks of length n beginning at the origin in the nearest-neighbour integer lattice Z d , and write Γ for a Wn-distributed walk. We show that the closing probability Wn ||Γn|| = 1 that Γ's endpoint neighbours the origin is at most n −1/2+o(1) in any dimension d ≥ 2. The method of proof is a reworking of that in [4], which found a closing probability upper bound of n −1/4+o(1) . A key element of the proof is made explicit and called the snake method. It is applied to prove the n −1/2+o(1) upper bound by means a technique of Gaussian pattern fluctuation.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On self-avoiding polygons and walks: The snake method via pattern fluctuation

Hammond

2019

Trans. Amer. Math. Soc.

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Self‐avoiding walks and polygons on hyperbolic graphs

Panagiotis

2024

Journal of Graph Theory

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We prove that for the ‐regular tessellations of the hyperbolic plane by ‐gons, there are exponentially more self‐avoiding walks of length than there are self‐avoiding polygons of length . We then prove that this property implies that the self‐avoiding walk is ballistic, even on an arbitrary vertex‐transitive graph. Moreover, for every fixed , we show that the connective constant for self‐avoiding walks satisfies the asymptotic expansion as ; 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 is comparable to the th power of their connective constant. Some of these results were previously obtained by Madras and Wu for all but finitely many regular tessellations of the hyperbolic plane.

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Self-Avoiding Walks and Connective Constants

Grimmett

2019

Springer Proceedings in Mathematics &Amp; Statistics

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The connective constant µ(G) of a quasi-transitive graph G is the asymptotic growth rate of the number of selfavoiding walks (SAWs) on G from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph G.• We present upper and lower bounds for µ in terms of the vertex-degree and girth of a transitive graph. • We discuss the question of whether µ ≥ φ for transitive cubic graphs (where φ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). • We present strict inequalities for the connective constants µ(G) of transitive graphs G, as G varies. • As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. • We describe so-called graph height functions within an account of 'bridges' for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function.• A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. • Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. • The review closes with a brief account of the 'speed' of SAW.

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On the probability that self-avoiding walk ends at a given point

Cited by 12 publications

References 22 publications

On self-avoiding polygons and walks: The snake method via pattern fluctuation

On self-avoiding polygons and walks: The snake method via pattern fluctuation

Self‐avoiding walks and polygons on hyperbolic graphs

Self-Avoiding Walks and Connective Constants

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