Spectral dimension of simple random walk on a long-range percolation cluster

Abstract

Consider the long-range percolation model on the integer lattice Zd in which all nearest-neighbour edges are present and otherwise x and y are connected with probability qx,y:=1−exp(−|x−y|−s), independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e. for s>d,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value d=1, s=2, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters s∈(d,2d). We further note that our approach is applicable to short-range models as well.