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March 2016 On the probability that self-avoiding walk ends at a given point
Hugo Duminil-Copin, Alexander Glazman, Alan Hammond, Ioan Manolescu
Ann. Probab. 44(2): 955-983 (March 2016). DOI: 10.1214/14-AOP993


We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on $\mathbb{Z}^{d}$ for $d\geq2$. We show that the probability that a walk of length $n$ ends at a point $x$ tends to $0$ as $n$ tends to infinity, uniformly in $x$. Also, when $x$ is fixed, with $\Vert x\Vert=1$, this probability decreases faster than $n^{-1/4+\varepsilon}$ for any $\varepsilon>0$. This provides a bound on the probability that a self-avoiding walk is a polygon.


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Hugo Duminil-Copin. Alexander Glazman. Alan Hammond. Ioan Manolescu. "On the probability that self-avoiding walk ends at a given point." Ann. Probab. 44 (2) 955 - 983, March 2016.


Received: 1 May 2013; Revised: 1 September 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1347.60131
MathSciNet: MR3474464
Digital Object Identifier: 10.1214/14-AOP993

Primary: 60K35
Secondary: 60D05

Keywords: endpoint delocalization , self-avoiding polygons , Self-avoiding walk

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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