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February 2014 Long-range last-passage percolation on the line
Sergey Foss, James B. Martin, Philipp Schmidt
Ann. Appl. Probab. 24(1): 198-234 (February 2014). DOI: 10.1214/13-AAP920

Abstract

We consider directed last-passage percolation on the random graph $G=(V,E)$ where $V=\mathbb{Z}$ and each edge $(i,j)$, for $i<j\in\mathbb{Z}$, is present in $E$ independently with some probability $p\in (0,1]$. To every $(i,j)\in E$ we attach i.i.d. random weights $v_{i,j}>0$. We are interested in the behaviour of $w_{0,n}$, which is the maximum weight of all directed paths from $0$ to $n$, as $n\rightarrow\infty$. We see two very different types of behaviour, depending on whether $\mathbb{E}[v_{i,j}^{2}]<\infty$ or $\mathbb{E}[v_{i,j}^{2}]=\infty$. In the case where $\mathbb{E}[v_{i,j}^{2}]<\infty$ we show that the process has a certain regenerative structure, and prove a strong law of large numbers and, under an extra assumption, a functional central limit theorem. In the situation where $\mathbb{E}[v_{i,j}^{2}]=\infty$ we obtain scaling laws and asymptotic distributions expressed in terms of a “continuous last-passage percolation” model on $[0,1]$; these are related to corresponding results for two-dimensional last-passage percolation with heavy-tailed weights obtained in Hambly and Martin [Probab. Theory Related Fields 137 (2007) 227–275].

Citation

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Sergey Foss. James B. Martin. Philipp Schmidt. "Long-range last-passage percolation on the line." Ann. Appl. Probab. 24 (1) 198 - 234, February 2014. https://doi.org/10.1214/13-AAP920

Information

Published: February 2014
First available in Project Euclid: 9 January 2014

zbMATH: 1293.60090
MathSciNet: MR3161646
Digital Object Identifier: 10.1214/13-AAP920

Subjects:
Primary: 60K35
Secondary: 05C80

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 1 • February 2014
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