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.
Publié le : 1978-06-14
Classification:
First passage percolation,
renewal theory,
subadditive processes,
60K05,
60F15,
94A20
@article{1176995525,
author = {Wierman, John C. and Reh, Wolfgang},
title = {On Conjectures in First Passage Percolation Theory},
journal = {Ann. Probab.},
volume = {6},
number = {6},
year = {1978},
pages = { 388-397},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995525}
}
Wierman, John C.; Reh, Wolfgang. On Conjectures in First Passage Percolation Theory. Ann. Probab., Tome 6 (1978) no. 6, pp. 388-397. http://gdmltest.u-ga.fr/item/1176995525/