April 2023 Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph
Sergey Foss, Takis Konstantopoulos, Artem Pyatkin
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Ann. Appl. Probab. 33(2): 931-953 (April 2023). DOI: 10.1214/22-AAP1832

Abstract

To each edge (i,j), i<j, of the complete directed graph on the integers we assign unit weight with probability p or weight x with probability 1p, independently from edge to edge, and give to each path weight equal to the sum of its edge weights. If W0,nx is the maximum weight of all paths from 0 to n then W0,nx/nCp(x), as n, almost surely, where Cp(x) is positive and deterministic. We study Cp(x) as a function of x, for fixed 0<p<1, and show that it is a strictly increasing convex function that is not differentiable if and only if x is a nonpositive rational or a positive integer except 1 or the reciprocal of it. We allow x to be any real number, even negative, or, possibly, . The case x= corresponds to the well-studied directed version of the Erdős–Rényi random graph (known as Barak–Erdős graph) for which Cp()=limxCp(x) has been studied as a function of p in a number of papers.

Funding Statement

The research of TK was partially supported by the CNRS PRC collaborative grant CNRS-193-382. The research of SF was partially supported by the Akademgorodok Mathematical Centre under agreement no. 075-15-2019-1675 with the Ministry of Science and Higher Education. The research of AP was supported by the Sobolev Institute of Mathematics contract no. 0314-2019-0014.

Acknowledgments

The authors would like to thank an anonymous referee who read the manuscript in great detail, made us aware of references [14] and [20] and suggested various improvements. The referee’s remarks helped us to largely improve the presentation of the paper, and, in particular, to provide a short proof for Lemma 11 and add all details of Proposition 9.

Citation

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Sergey Foss. Takis Konstantopoulos. Artem Pyatkin. "Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph." Ann. Appl. Probab. 33 (2) 931 - 953, April 2023. https://doi.org/10.1214/22-AAP1832

Information

Received: 1 June 2020; Revised: 1 March 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 07692280
MathSciNet: MR4564417
Digital Object Identifier: 10.1214/22-AAP1832

Subjects:
Primary: 05C80 , 60K05 , 60K35

Keywords: critical point , Last-passage percolation , maximal path , random graph , Regenerative structure , skeleton point

Rights: Copyright © 2023 Institute of Mathematical Statistics

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Vol.33 • No. 2 • April 2023
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