The Annals of Applied Probability

Long-range last-passage percolation on the line

Sergey Foss, James B. Martin, and Philipp Schmidt

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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].

Article information

Source
Ann. Appl. Probab., Volume 24, Number 1 (2014), 198-234.

Dates
First available in Project Euclid: 9 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1389278724

Digital Object Identifier
doi:10.1214/13-AAP920

Mathematical Reviews number (MathSciNet)
MR3161646

Zentralblatt MATH identifier
1293.60090

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05C80: Random graphs [See also 60B20]

Keywords
Last-passage percolation directed random graph regenerative structure regular variation heavy tails

Citation

Foss, Sergey; Martin, James B.; Schmidt, Philipp. Long-range last-passage percolation on the line. Ann. Appl. Probab. 24 (2014), no. 1, 198--234. doi:10.1214/13-AAP920. https://projecteuclid.org/euclid.aoap/1389278724


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