Summary
We consider random walk on the infinite cluster of bond percolation on ℤd. We show that, in the supercritical regime whend≧3, this random walk is a.s. transient. This conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance. It is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite.
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G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University
H.K. was supported in part by the N.S.F. through a grant to Cornell University
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Grimmett, G.R., Kesten, H. & Zhang, Y. Random walk on the infinite cluster of the percolation model. Probab. Th. Rel. Fields 96, 33–44 (1993). https://doi.org/10.1007/BF01195881
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DOI: https://doi.org/10.1007/BF01195881