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M.T. Barlow

Random walks on percolation clusters

Abstract: We consider the simple random walk on the infinite cluster in supercritical bond percolation in $Z^d$. Using 'heat kernel' techniques ultimately based on ideas of Nash, one can obtain Gaussian upper and lower bounds on the transition probabilities. These estimates have a number of easy consequences: an (annealed) invariance principle, and the Liouville property for positive harmonic functions.