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First-passage percolation on the square lattice. I

Published online by Cambridge University Press:  01 July 2016

R. T. Smythe  and
John C. Wierman
R. T. Smythe*
Affiliation:
University of Washington
John C. Wierman*
Affiliation:
University of Washington
*
∗Now at the University of Oregon. Research supported in part by NSF Grant MPS 74–07424 A01.
∗∗Now at the University of Minnesota.
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Abstract

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

Keywords

FIRST-PASSAGE PERCOLATIONRENEWAL THEORYSUBADDITIVE PROCESSESPERCOLATION PROBABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 1 , March 1977 , pp. 38 - 54
Copyright
Copyright © Applied Probability Trust 1977 

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References

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