## Journal of Mathematics of Kyoto University

- J. Math. Kyoto Univ.
- Volume 46, Number 4 (2006), 903-912.

### A note on anisotropic first-passage percolation

#### Abstract

We consider a first-passage percolation problem on the square lattice, where the distribution function of time coordinates of horizontal edges may be different from that of vertical edges. Some basic limit theorems for first-passage times and minimal lengths of optimal paths are obtained. Especially, we show that as long as the system is in the supercritical phase, the expectation of first-passage time from the origin to a point with distance $n$ converges to a finite constant, which is independent of the directions, as $n \to \infty$.

#### Article information

**Source**

J. Math. Kyoto Univ., Volume 46, Number 4 (2006), 903-912.

**Dates**

First available in Project Euclid: 14 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1250281609

**Digital Object Identifier**

doi:10.1215/kjm/1250281609

**Mathematical Reviews number (MathSciNet)**

MR2320356

**Zentralblatt MATH identifier**

1137.60348

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

#### Citation

Takei, Masato. A note on anisotropic first-passage percolation. J. Math. Kyoto Univ. 46 (2006), no. 4, 903--912. doi:10.1215/kjm/1250281609. https://projecteuclid.org/euclid.kjm/1250281609