## The Annals of Applied Probability

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

### Long-range last-passage percolation on the line

Sergey Foss, James B. Martin, and Philipp Schmidt

#### 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