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November 2003 Large deviations in first-passage percolation
Yunshyong Chow, Yu Zhang
Ann. Appl. Probab. 13(4): 1601-1614 (November 2003). DOI: 10.1214/aoap/1069786513

Abstract

Consider the standard first-passage percolation on ${\Z}^d$, $d\geq 2$. Denote by $\phi_{0,n}$ the face--face first-passage time in $[0,n]^d$. It is well known that \[ \lim_{n\rightarrow \infty} {\phi_{0,n}\over n}=\mu(F) \qquad \mbox{a.s. and in } L_1, \] where $F$ is the common distribution on each edge. In this paper we show that the upper and lower tails of $\phi_{0,n}$ are quite different when $\mu(F)>0$. More precisely, we can show that for small $\varepsilon>0$, there exist constants $\alpha(\varepsilon, F)$ and $\beta (\varepsilon, F)$ such that \[ \lim_{n\rightarrow\infty}{-1\over n} \log P \left( \phi_{0,n}\leq n(\mu -\varepsilon) \right) = \alpha (\varepsilon, F) \] and \[ \lim_{n\rightarrow\infty}{-1\over n^d} \log P \left(\phi_{0,n}\geq n(\mu +\varepsilon) \right)= \beta (\varepsilon, F). \]

Citation

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Yunshyong Chow. Yu Zhang. "Large deviations in first-passage percolation." Ann. Appl. Probab. 13 (4) 1601 - 1614, November 2003. https://doi.org/10.1214/aoap/1069786513

Information

Published: November 2003
First available in Project Euclid: 25 November 2003

zbMATH: 1038.60093
MathSciNet: MR2023891
Digital Object Identifier: 10.1214/aoap/1069786513

Subjects:
Primary: 60K35

Keywords: First-passage percolation , large deviations

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.13 • No. 4 • November 2003
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