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

Author:

Hammond Alan

Abstract

For d 2 d \geq 2 and n N n \in \mathbb {N} , let W n \mathsf {W}_n denote the uniform law on self-avoiding walks of length  n n beginning at the origin in the nearest-neighbour integer lattice  Z d \mathbb {Z}^d , and write Γ \Gamma for a W n \mathsf {W}_n -distributed walk. We show that in the closing probability W n ( | | Γ n | | = 1 ) \mathsf {W}_n \big ( \vert \vert \Gamma _n \vert \vert = 1 \big ) that Γ \Gamma ’s endpoint neighbours the origin and is at most n 1 / 2 + o ( 1 ) n^{-1/2 + o(1)} in any dimension d 2 d \geq 2 . The method of proof is a reworking of that in [Ann. Probab. 44 (2016), pp. 955–983], which found a closing probability upper bound of n 1 / 4 + o ( 1 ) 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 ) n^{-1/2 + o(1)} upper bound by means of a technique of Gaussian pattern fluctuation.

Funder

National Science Foundation

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference22 articles.

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