Published online by Cambridge University Press: 01 July 2016
The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.
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