Last passage percolation on the complete graph

Feng Wang, Xian-Yuan Wu and Rui Zhu

Statistics & Probability Letters, 2020, vol. 164, issue C

Abstract: We consider the last passage percolation on the complete graph. Let Gn=([n],En) be the complete graph on vertex set [n]={1,2,…,n}, and i.i.d. sequence {Xe:e∈En} be the passage times of edges. Denote by Wn the largest passage time among all self-avoiding paths from 1 to n. First, it is proved that Wn∕n converges to constant μ, where μ is called the time constant and coincides with the essential supremum of Xe. Second, when μ=∞, lower and upper bounds for Wn∕n are given, in particular, for a large class of passage times with light tails, a weak law of large number for Wn is obtained. Finally, when μ<∞, it is proved that the deviation probability P(Wn∕n≤μ−x) decays as fast as e−Θ(n2).

Keywords: Last passage percolation; Time constant; Random graph; Law of large number; Deviation probability; Depth-First-Search algorithm (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spl.2020.108798

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