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

Consider the long-range percolation model on the integer lattice Z^d in which all nearest-neighbour edges are present and otherwise x and y are connected with probability q_x,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.