## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 6, Number 3 (1978), 388-397.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995525

Digital Object Identifier
doi:10.1214/aop/1176995525

Mathematical Reviews number (MathSciNet)
MR478390

Zentralblatt MATH identifier
0394.60084

JSTOR
links.jstor.org

Subjects
Primary: 60K05: Renewal theory
Secondary: 60F15: Strong theorems 94A20: Sampling theory

#### Citation

Wierman, John C.; Reh, Wolfgang. On Conjectures in First Passage Percolation Theory. Ann. Probab. 6 (1978), no. 3, 388--397. doi:10.1214/aop/1176995525. https://projecteuclid.org/euclid.aop/1176995525