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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=Z and each edge (i,j), for i<jZ, is present in E independently with some probability p(0,1]. To every (i,j)E we attach i.i.d. random weights vi,j>0. We are interested in the behaviour of w0,n, which is the maximum weight of all directed paths from 0 to n, as n. We see two very different types of behaviour, depending on whether E[vi,j2]< or E[vi,j2]=. In the case where E[vi,j2]< 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 E[vi,j2]= 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

Keywords: directed random graph , heavy tails , Last-passage percolation , Regenerative structure , regular variation

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.24 • No. 1 • February 2014
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