The Annals of Probability

Large deviations for the chemical distance in supercritical Bernoulli percolation

Olivier Garet and Régine Marchand

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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.

Article information

Source
Ann. Probab., Volume 35, Number 3 (2007), 833-866.

Dates
First available in Project Euclid: 10 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1178804316

Digital Object Identifier
doi:10.1214/009117906000000881

Mathematical Reviews number (MathSciNet)
MR2319709

Zentralblatt MATH identifier
1117.60090

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Percolation first-passage percolation chemical distance shape theorem large deviation inequalities

Citation

Garet, Olivier; Marchand, Régine. Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 35 (2007), no. 3, 833--866. doi:10.1214/009117906000000881. https://projecteuclid.org/euclid.aop/1178804316


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