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 ≥ 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 \({t^{-{d}/_{s-d}}}\) up to logarithmic corrections, where t denotes the time the walk is run. Our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. The bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper (Crawford and Sly in Simple randomwalk on long range percolation clusters II: scaling limit, 2010) where we establish the scaling limit of the random walk to be α-stable Lévy motion.
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N. Crawford supported in part at the Technion by an Marilyn and Michael Winer Fellowship. Work completed while the authors were at UC Berkeley Department of Statistics.
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Crawford, N., Sly, A. Simple random walk on long range percolation clusters I: heat kernel bounds. Probab. Theory Relat. Fields 154, 753–786 (2012). https://doi.org/10.1007/s00440-011-0383-2
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DOI: https://doi.org/10.1007/s00440-011-0383-2