## The Annals of Applied Probability

### Large deviations in first-passage percolation

#### 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 $\lim_{n\rightarrow\infty}{-1\over n^d} \log P \left(\phi_{0,n}\geq n(\mu +\varepsilon) \right)= \beta (\varepsilon, F).$

#### Article information

Source
Ann. Appl. Probab., Volume 13, Number 4 (2003), 1601-1614.

Dates
First available in Project Euclid: 25 November 2003

https://projecteuclid.org/euclid.aoap/1069786513

Digital Object Identifier
doi:10.1214/aoap/1069786513

Mathematical Reviews number (MathSciNet)
MR2023891

Zentralblatt MATH identifier
1038.60093

#### Citation

Chow, Yunshyong; Zhang, Yu. Large deviations in first-passage percolation. Ann. Appl. Probab. 13 (2003), no. 4, 1601--1614. doi:10.1214/aoap/1069786513. https://projecteuclid.org/euclid.aoap/1069786513

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