Heat Kernel Upper Bounds on Long Range Percolation Clusters

Abstract

In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any $d \geq 1$ and for any exponent $s \in (d, (d+2) \wedge 2d)$ giving the rate of decay of the percolation process, we show that the return probability decays like $t^{-\ffrac{d}{s-d}}$ up to logarithmic corrections, where $t$ denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest, the main result is needed for a companion paper studying the scaling limit of simple random walk on the infinite cluster.