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First-Passage Percolation on the Square Lattice. I

R. T. Smythe and John C. Wierman
Advances in Applied Probability
Vol. 9, No. 1 (Mar., 1977), pp. 38-54
DOI: 10.2307/1425815
Stable URL: http://www.jstor.org/stable/1425815
Page Count: 17
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
First-Passage Percolation on the Square Lattice. I
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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.

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