Return probabilities of a simple random walk on percolation clusters
Denote by \(\mathcal C_\infty(\mathbb Z^d,p)\) the infinite percolation cluster obtained by a (supercritical) Bernoulli bond percolation on \(\mathbb Z^d\) where bonds are retained with probability \(p\) and removed with probability \(1-p\). The authors study simple random walks (SRWs) on \(\mathcal C_\infty(\mathbb Z^d,p)\). It is well-known from papers by \textit{G. R. Grimmett, H. Kesten} and \textit{Y. Zhang} [Probab. Theory Relat. Fields 96, No. 1, 33--44 (1993; Zbl 0791.60095)] that in dimensions \(d\geq 3\) a SRW is almost surely transient. The authors prove that \(C' t^{-d/2} (\log t)^{6d+14}\) is an upper bound for the probability that a SRW returns to the origin at time \(t\). (This compares with a lower bound of \(C t^{-d/2}/\log t\) due to \textit{N. T. Varopoulos} [Bull. Sci. Math., II. Sér. 109, 225--252 (1985; Zbl 0583.60063)] and \textit{T. K. Carne} [ibid. 109, 399--405 (1985; Zbl 0584.60078)].) As an application the authors show that in dimensions 5 and higher the uniform spanning forest on infinite percolation clusters is supported on infinitely many components. Some of the results of this paper are now superseded by recent progress due to \textit{M. T. Barlow} [Ann. Probab. 32, No. 4, 3024--3084 (2004; Zbl 1067.60101)].
scientific article; zbMATH DE number 2142162