On the probability that self-avoiding walk ends at a given point (Q282496)

This paper considers the uniform distribution of self-avoiding walks in \(\mathbb{Z}^d\) of length \(n\) and looks at the decay probability that the walk ends at a given point. Two important theorems are proved for \(d \geq 2\). The first is that the probability that the end point of the walk lies at distance one from the origin is bounded above by \(n^{-1/4}\) for large \(n\). The second is that the probability that the walk ends at a fixed \(x \in \mathbb{Z}^d\) tends to zero as \(n \to \infty\). The inequality in the first theorem is not believed to be sharp.NEWLINENEWLINESince the self-avoiding walk is a hard combinatorial object, even seemingly obvious statements such as the ones above do not seem to have an easy proof. An important ingredient in the proofs are two ideas: a pattern theorem due to \textit{H. Kesten} [J. Math. Phys. 4, 960--969 (1963; Zbl 0122.36502)], and an unfolding argument due to \textit{J. M. Hammersley} and \textit{D. J. A. Welsh} [Q. J. Math., Oxf. II. Ser. 13, 108--110 (1962; Zbl 0123.00304)].