Statistics Papers

Document Type

Journal Article

Date of this Version

2003

Publication Source

The Annals of Probability

Volume

31

Issue

2

Start Page

1028

Last Page

1051

DOI

10.1214/aop/1048516544

Abstract

We study the paths of minimal cost for first-passage percolation in two dimensions and obtain an exponential bound on the tail probability of the ratio of the lengths of the shortest and longest of these. This inequality permits us to answer a long-standing question of Hammersley and Welsh (1965) on the shift differentiability of the time constant. Specifically, we show that for subcritical Bernoulli percolation the time constant is not shift differentiable when p is close to one-half.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aop/1048516544#info

Keywords

first-passage percolation, Bernoulli percolation, Hammersley, Erlsh, differentiability, time constants, shortest path, longest path, surgery

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Date Posted: 27 November 2017

This document has been peer reviewed.