Maximal edge-traversal time in First Passage Percolation
Shuta Nakajima
TL;DR
This paper investigates the growth rate of the maximal edge-traversal time on optimal paths in first passage percolation, providing bounds for various distributions including Pareto and Weibull.
Contribution
It determines the order of growth of the maximal weight in first passage percolation for multiple edge distributions, extending known unboundedness results.
Findings
Maximal weight growth is unbounded for distributions with unbounded support.
The order of growth of the maximal weight is established up to a multiplicative constant.
Results apply to Pareto and Weibull distributions, among others.
Abstract
In this paper, we study the maximal edge-traversal time (simply we call maximal weight hereafter) on the optimal paths in the first passage percolation for several edge distributions, including the Pareto and Weibull distributions. It is known to be unbounded when the edge distribution has unbounded support [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993]. We determine the order of the growth up to a multiplicative constant.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Random Matrices and Applications · Markov Chains and Monte Carlo Methods