Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph Full article
| Journal |
Annals of Applied Probability
ISSN: 1050-5164 |
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| Output data | Year: 2023, Volume: 33, Number: 2, Pages: 731-753 Pages count : 23 DOI: 10.1214/22-aap1832 | ||||||
| Tags | critical point, Last-passage percolation, maximal path, random graph, Regenerative structure, skeleton point | ||||||
| Authors |
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| Affiliations |
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Funding (2)
| 1 | Sobolev Institute of Mathematics | 0314-2019-0014 |
| 2 | Mathematical Center in Akademgorodok | 075-15-2019-1675 |
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 1 − p, independently from edge to edge, and give to each path weight equal to the sum of its edge weights. If W0x,n is the maximum weight of all paths from 0 to n then W0x,n/n → Cp(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(−∞) = limx→−∞ Cp(x) has been studied as a function of p in a number of papers.
Cite:
Foss S.
, Konstantopoulos T.
, Pyatkin A.
Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph
Annals of Applied Probability. 2023. V.33. N2. P.731-753. DOI: 10.1214/22-aap1832 WOS Scopus РИНЦ OpenAlex
Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph
Annals of Applied Probability. 2023. V.33. N2. P.731-753. DOI: 10.1214/22-aap1832 WOS Scopus РИНЦ OpenAlex
Dates:
| Submitted: | Jun 1, 2020 |
| Published print: | Mar 21, 2023 |
| Published online: | Apr 1, 2023 |
Identifiers:
| Web of science: | WOS:000960867800004 |
| Scopus: | 2-s2.0-85152898137 |
| Elibrary: | 61242636 |
| OpenAlex: | W3033847703 |