Open Access
May, 1993 On the Speed of Convergence in First-Passage Percolation
Harry Kesten
Ann. Appl. Probab. 3(2): 296-338 (May, 1993). DOI: 10.1214/aoap/1177005426

Abstract

We consider the standard first-passage percolation problem on Zd:{t(e):ean edge ofZd} is an i.i.d. family of random variables with common distribution F,a0,n:=inf{1kt(e1):(e1,,ek) a path on Zd from 0 to nξ1}, where ξ1 is the first coordinate vector. We show that σ2(a0,n)C1n and that P{|a0,nEa0,n|xn}C2exp(C3x) for xC4n and for some constants 0<Ci<. It is known that μ:=lim(1/n)Ea0,n exists. We show also that C5n1Ea0,nnμC6n5/6(logn)1/3. There are corresponding statements for the roughness of the boundary of the set B~(t)={ν:ν a vertex of Zd that can be reached from the origin by a path (e1,,ek) with t(ei)t}.

Citation

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Harry Kesten. "On the Speed of Convergence in First-Passage Percolation." Ann. Appl. Probab. 3 (2) 296 - 338, May, 1993. https://doi.org/10.1214/aoap/1177005426

Information

Published: May, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0783.60103
MathSciNet: MR1221154
Digital Object Identifier: 10.1214/aoap/1177005426

Subjects:
Primary: 60K35
Secondary: 60F05 , 60F10

Keywords: asymptotic shape , Eden model , First-passage percolation , method of bounded differences , roughness of boundary , Speed of convergence

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.3 • No. 2 • May, 1993
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