On self-avoiding polygons and walks: The snake method via pattern fluctuation
Author:
Alan Hammond
Journal:
Trans. Amer. Math. Soc. 372 (2019), 2335-2356
MSC (2010):
Primary 60K35; Secondary 60D05
DOI:
https://doi.org/10.1090/tran/7494
Published electronically:
May 23, 2019
MathSciNet review:
3988578
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For $d \geq 2$ and $n \in \mathbb {N}$, let $\mathsf {W}_n$ denote the uniform law on self-avoiding walks of length $n$ beginning at the origin in the nearest-neighbour integer lattice $\mathbb {Z}^d$, and write $\Gamma$ for a $\mathsf {W}_n$-distributed walk. We show that in the closing probability $\mathsf {W}_n \big ( \vert \vert \Gamma _n \vert \vert = 1 \big )$ that $\Gamma$’s endpoint neighbours the origin and is at most $n^{-1/2 + o(1)}$ in any dimension $d \geq 2$. 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 $n^{-1/4 + o(1)}$. 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 of a technique of Gaussian pattern fluctuation.
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Additional Information
Alan Hammond
Affiliation:
Departments of Mathematics and Statistics, University of California Berkeley, 899 Evans Hall, Berkeley, California 94720-3840
MR Author ID:
753771
Email:
alanmh@stat.berkeley.edu
Received by editor(s):
April 7, 2017
Received by editor(s) in revised form:
October 23, 2017
Published electronically:
May 23, 2019
Additional Notes:
The author was supported by NSF grant DMS-1512908.
Article copyright:
© Copyright 2019
Alan Hammond