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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 $\mathbb{Z}^d: \{t(e): e \text{an edge of} \mathbb{Z}^d\}$ is an i.i.d. family of random variables with common distribution $F, a_{0,n} := \inf\{\sum^k_1 t(e_1): (e_1, \cdots, e_k)$ a path on $\mathbb{Z}^d$ from 0 to $n \xi_1\}$, where $\xi_1$ is the first coordinate vector. We show that $\sigma^2(a_{0,n}) \leq C_1 n$ and that $P\{|a_{0,n} - Ea_{0,n}| \geq x\sqrt{n}\} \leq C_2 \exp(-C_3 x)$ for $x \leq C_4 n$ and for some constants $0 < C_i < \infty$. It is known that $\mu := \lim(1/n)Ea_{0,n}$ exists. We show also that $C_5 n^{-1} \leq Ea_{0,n} - n\mu \leq C_6 n^{5/6}(\log n)^{1/3}$. There are corresponding statements for the roughness of the boundary of the set $\tilde{B}(t) = \{\nu: \nu$ a vertex of $\mathbb{Z}^d$ that can be reached from the origin by a path $(e_1, \cdots, e_k)$ with $\sum t(e_i) \leq t\}$.

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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

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Vol.3 • No. 2 • May, 1993
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