On Conjectures in First Passage Percolation Theory

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.