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2019 On self-avoiding polygons and walks: the snake method via polygon joining
Alan Hammond
Electron. J. Probab. 24: 1-43 (2019). DOI: 10.1214/18-EJP249


For $d \geq 2$ and $n \in \mathbb{N} $, let $\mathsf{W} _n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z} ^d$, and write $\Gamma $ for a $\mathsf{W} _n$-distributed walk. We show that the closing probability $\mathsf{W} _n \big (\vert \vert \Gamma _n \vert \vert = 1 \big )$ that $\Gamma $’s endpoint neighbours the origin is at most $n^{-4/7 + o(1)}$ for a positive density set of odd $n$ in dimension $d = 2$. This result is proved using the snake method, a general technique for proving closing probability upper bounds, which originated in [3] and was made explicit in [8]. Our conclusion is reached by applying the snake method in unison with a polygon joining technique whose use was initiated by Madras in [13].


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Alan Hammond. "On self-avoiding polygons and walks: the snake method via polygon joining." Electron. J. Probab. 24 1 - 43, 2019.


Received: 10 January 2018; Accepted: 20 November 2018; Published: 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07068780
MathSciNet: MR3954789
Digital Object Identifier: 10.1214/18-EJP249

Primary: 60K35
Secondary: 60D05

Keywords: combinatorial bounds , planar self-avoiding walk , return probability


Vol.24 • 2019
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