The Annals of Probability

Isoperimetry and heat kernel decay on percolation clusters

Pierre Mathieu and Elisabeth Remy

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We prove that the heat kernel on the infinite Bernoulli percolation cluster in $\Z^d$ almost surely decays faster than $t^{-d/2}$. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities. Some of the results of this paper were previously announced in the note of Mathieu and Remy [C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 927--931].

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Ann. Probab., Volume 32, Number 1A (2004), 100-128.

First available in Project Euclid: 4 March 2004

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Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Percolation isoperimetry spectral gap heat kernel decay


Mathieu, Pierre; Remy, Elisabeth. Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32 (2004), no. 1A, 100--128. doi:10.1214/aop/1078415830.

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