Supercritical behaviors in first-passage percolation
type: Item
instance of: scholarly article
title: Supercritical behaviors in first-passage percolation
zbMATH Open document ID: 0840.60090
DOI: 10.1016/0304-4149(95)00051-8
published in: Stochastic Processes and their Applications
publication date: 1996-06-30T00:00:00Z
review text: The paper obtains some nice results on the existence of routes in standard first passage percolation on \(Z^d\) where the nonnegative random variables \(X(e)\) \((e\) an edge of \(Z^d)\) are i.i.d. with distribution function \(F\). Let \(a_{0,n}\) denote the first passage time from the origin to \((n,0, \dots,0)\), and denote by \(b_{0,n}\) the first passage time from the origin to the set \(\{(n, k_2, \dots, k_d) \in Z^d : k_2, \dots, k_d \in Z\}\). A path \(\gamma\) is called a route for \(a_{0,n}\) \((b_{0,n})\) if the passage time of \(\gamma\) equals \(a_{0,n}\) \((b_{0, n})\). It is shown in the case \(F(0) > p_c(d)\) (where \(p_c (d)\) denotes the critical probability for Bernoulli (bond) percolation on \(Z^d)\): There exist routes for both \(a_{0,n}\) and \(b_{0,n}\). On the other hand there exists a random variable \(\rho (F) \geq 0\) such that \(E [(\rho (F))^k] < \infty\) if \(E [(X(e))^k] < \infty\) for some \(k \geq 1\), and, as \(n \to \infty\), \(E [a_{0,n}] \to 2E [\rho (F)]\), \(E [b_{0,n}] \to E [\rho(F)]\).
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 828753
zbMATH Keywords: first passage percolation
zbMATH Keywords: first passage time
author: Y. Zhang
reviewed by: Klaus Schürger
MaRDI profile type: Publication
full work available at URL: 0304-4149(95)00051-8
OpenAlex ID: W1966888298
cites work: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation
cites work: Q4067293
cites work: Critical behavior of the two-dimensional first passage time
cites work: Q4693063
cites work: The supercritical phase of percolation is well behaved
cites work: Q3738376
cites work: The probability of a large cluster in supercritical Bernoulli perlocation
cites work: A shape theorem for epidemics and forest fires with finite range interactions
cites work: A limit theorem for \(N_{0n}/n\) in first-passage percolation
description: scientific article; zbMATH DE number 828753