Nondifferentiability of the time constants of first-passage percolation
type: Item
instance of: scholarly article
title: Nondifferentiability of the time constants of first-passage percolation
zbMATH Open document ID: 1029.82017
author: J. Michael Steele
author: Y. Zhang
published in: The Annals of Probability
publication date: 2004-02-05T00:00:00Z
review text: This paper studies the time constant in the first-passage percolation (FPP) problem on \(\mathbb{Z}^2\), introduced in 1965 by Hammersley and Welsh, and answer negatively to an open conjecture of this field. The model studied here on \(\mathbb{Z}^2\) is this of the general FPP problem but the main result concerns the Bernoulli case. On the lattice \(\mathbb{Z}^2\), one considers a random field \(x=(x(e))_{e \in\mathbb{Z}^2}\) of i.i.d. random variables, with a marginal distribution \(F\) of finite mean. When \(e=(u,v)\), this random variable \(x(e)\) can be viewed as the amount of time needed to go from \(u\) to \(v\). In the simplest Bernoulli case, this amount is either \(0\) or \(1\) with probability \(p\) and \(1-p\). To study the stream along the lattice, one also considers paths along edges \(\gamma = \{v_0,e_1,v_1,\dots,e_n,v_n\}\) and defines the passage time of \(\gamma\) to be \(\tau(\gamma)=\sum_{e \in \gamma} x(e)\). Hammersley and Welsh had first studied the first passage time (FPT) \(a_{0,n}\) from the origin and the point \(u(n,0)\) of the horizontal axis, and proved by subbadditivity that there exists a finite constant \(\mu(F)\) such that the ratio \(a_{0,n}/n\) converges (a.s. and in \(L^1\)) to \(F\) when \(n\) goes to infinity. This so-called time constant \(\mu(F)\) has been studied intensively and is a central object of this paper. Depending on the configuration, different paths can lead to the FPT \(a_{0,n}\) (these paths are then called routes. Another object of interest, because it provides information on the FPT, is the length \(N_n\) of the shortest route. This length behaves differently depending on the percolation regime of the model. In the supercritical case, Zhang and Zhang (1984) proved that \(N_n/n\) converges a.s. and in \(L^1\) to another finite time constant \(\lambda(F)\), but this problem is still open in other regimes. Hammersley and Welsh had already studied this question and transformed it into a problem of smoothness of the…
reviewed by: Arnaud Le Ny
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 1933039
zbMATH Keywords: first-passage percolation
zbMATH Keywords: Bernoulli percolation
zbMATH Keywords: Hammersley Welsh time constants
zbMATH Keywords: shortest path
zbMATH Keywords: longest path
zbMATH Keywords: surgery
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1048516544
OpenAlex ID: W2013248203
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DOI: 10.1214/AOP/1048516544
description: scientific article; zbMATH DE number 1933039