Heat Kernel Upper Bounds on Long Range Percolation Clusters

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 ≥ 1 and for any exponent s ∈ (d, (d + 2) ∧ 2d) giving the rate of decay of the percolation process, we show that the return probability decays like t− /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.