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June, 1978 On Conjectures in First Passage Percolation Theory
John C. Wierman, Wolfgang Reh
Ann. Probab. 6(3): 388-397 (June, 1978). DOI: 10.1214/aop/1176995525

Abstract

We consider several conjectures of Hammersley and Welsh in the theory of first passage percolation on the two-dimensional rectangular lattice. Our results include: (i) a proof that the time constant is zero when the atom at zero of the underlying distribution is one-half or larger; (ii) almost sure existence of routes for the unrestricted first passage times; (iii) almost sure limit theorems for the first passages $s_{0n}$ and $b_{0n}$, the reach processes $y_t$ and $y^u_t$, and the route length processes $N^s_n$ and $N^b_n$; (iv) bounds on the expected maximum height of routes for $s_{0n}$ and $t_{0n}$ when the atom at zero of the underlying distribution is one-half or larger.

Citation

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John C. Wierman. Wolfgang Reh. "On Conjectures in First Passage Percolation Theory." Ann. Probab. 6 (3) 388 - 397, June, 1978. https://doi.org/10.1214/aop/1176995525

Information

Published: June, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0394.60084
MathSciNet: MR478390
Digital Object Identifier: 10.1214/aop/1176995525

Subjects:
Primary: 60K05
Secondary: 60F15, 94A20

Rights: Copyright © 1978 Institute of Mathematical Statistics

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Vol.6 • No. 3 • June, 1978
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