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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: May 23, 2019
MathSciNet review: 3988578
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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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Alan Hammond
Affiliation: Departments of Mathematics and Statistics, University of California Berkeley, 899 Evans Hall, Berkeley, California 94720-3840

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