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Random walks on supercritical percolation clusters. (English) Zbl 1067.60101
Gaussian upper and lower bounds are obtained on the transition density of the continuous time simple random walk on a supercritical percolation cluster in the Euclidean space. The paper begins from a resume of random walks on graphs. Next, percolation estimates are established and Poincaré inequalities are derived. For a general graph that satisfies approporiate volume growth and Poincaré inequalities, the two-sided bound is derived for the transition density. These bounds are analogous to Aronsen’s bounds for uniformly elliptic divergence from diffusions and depend on the percolation probability. The irregular nature of the medium means that the bound for the transition density holds only for times exceeding certain percolation configuration-dependent value.

MSC:
60K37 Processes in random environments
58J35 Heat and other parabolic equation methods for PDEs on manifolds
82B43 Percolation
05C80 Random graphs (graph-theoretic aspects)
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