## The Annals of Probability

### On the probability that self-avoiding walk ends at a given point

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 44, Number 2 (2016), 955-983.

Dates
Revised: September 2014
First available in Project Euclid: 14 March 2016

https://projecteuclid.org/euclid.aop/1457960388

Digital Object Identifier
doi:10.1214/14-AOP993

Mathematical Reviews number (MathSciNet)
MR3474464

Zentralblatt MATH identifier
1347.60131

#### Citation

Duminil-Copin, Hugo; Glazman, Alexander; Hammond, Alan; Manolescu, Ioan. On the probability that self-avoiding walk ends at a given point. Ann. Probab. 44 (2016), no. 2, 955--983. doi:10.1214/14-AOP993. https://projecteuclid.org/euclid.aop/1457960388

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