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May 2007 Large deviations for the chemical distance in supercritical Bernoulli percolation
Olivier Garet, Régine Marchand
Ann. Probab. 35(3): 833-866 (May 2007). DOI: 10.1214/009117906000000881

Abstract

The chemical distance D(x, y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behavior of this random metric, and we prove that, for an appropriate norm μ depending on the dimension and the percolation parameter, the probability of the event $$\biggl\{\ 0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\varepsilon ,1+\varepsilon )\ \biggr\}$$ exponentially decreases when ‖x1 tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.

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Olivier Garet. Régine Marchand. "Large deviations for the chemical distance in supercritical Bernoulli percolation." Ann. Probab. 35 (3) 833 - 866, May 2007. https://doi.org/10.1214/009117906000000881

Information

Published: May 2007
First available in Project Euclid: 10 May 2007

zbMATH: 1117.60090
MathSciNet: MR2319709
Digital Object Identifier: 10.1214/009117906000000881

Subjects:
Primary: 60K35
Secondary: 82B43

Keywords: Chemical distance , First-passage percolation , large deviation inequalities , percolation , shape theorem

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 3 • May 2007
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