Nondifferentiability of Time Constants for First-Passage Percolation
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Statistics Papers
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first-passage percolation
Bernoulli percolation
Hammersley
Erlsh
differentiability
time constants
shortest path
longest path
surgery
Physical Sciences and Mathematics
Bernoulli percolation
Hammersley
Erlsh
differentiability
time constants
shortest path
longest path
surgery
Physical Sciences and Mathematics
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Steele, J. Michael
Zhang, Yu
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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.
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2003-01-01
Journal title
The Annals of Probability