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April 2003 Nondifferentiability of the time constants of first-passage percolation
J. Michael Steele, Yu Zhang
Ann. Probab. 31(2): 1028-1051 (April 2003). 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 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.

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J. Michael Steele. Yu Zhang. "Nondifferentiability of the time constants of first-passage percolation." Ann. Probab. 31 (2) 1028 - 1051, April 2003. https://doi.org/10.1214/aop/1048516544

Information

Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1029.82017
MathSciNet: MR1964957
Digital Object Identifier: 10.1214/aop/1048516544

Subjects:
Primary: 82B43
Secondary: 60K35

Keywords: Bernoulli percolation , differentiability , First-passage percolation , Hammersley , longest path , Shortest path , surgery , time constants , Welsh

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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