## The Annals of Probability

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

### On Conjectures in First Passage Percolation Theory

John C. Wierman and Wolfgang Reh

#### 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

**Keywords**

First passage percolation renewal theory subadditive processes

#### 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