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- W2087890393 abstract "We consider directed last-passage percolation on the random graph $G=(V,E)$ where $V=mathbb{Z}$ and each edge $(i,j)$, for $i<jinmathbb{Z}$, is present in $E$ independently with some probability $pin (0,1]$. 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 $nrightarrowinfty$. 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]." @default.
- W2087890393 created "2016-06-24" @default.
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- W2087890393 date "2014-02-01" @default.
- W2087890393 modified "2023-10-12" @default.
- W2087890393 title "Long-range last-passage percolation on the line" @default.
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- W2087890393 doi "https://doi.org/10.1214/13-aap920" @default.
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