## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 21, Number (2016), paper no. 24, 23 pp.

### On the time constant of high dimensional first passage percolation

#### Abstract

We study the time constant $\mu (e_{1})$ in first passage percolation on $\mathbb Z^{d}$ as a function of the dimension. We prove that if the passage times have finite mean, $$\lim_{d\to\infty}\frac{\mu({e}_{1})d}{\log d} = \frac{1}{2a},$$ where $a \in [0,\infty ]$ is a constant that depends only on the behavior of the distribution of the passage times at $0$. For the same class of distributions, we also prove that the limit shape is not an Euclidean ball, nor a $d$-dimensional cube or diamond, provided that $d$ is large enough.

#### Article information

**Source**

Electron. J. Probab. Volume 21, Number (2016), paper no. 24, 23 pp.

**Dates**

Received: 11 February 2016

Accepted: 15 March 2016

First available in Project Euclid: 5 April 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1459880112

**Digital Object Identifier**

doi:10.1214/16-EJP1

**Zentralblatt MATH identifier**

1338.60227

**Subjects**

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

**Keywords**

first passage percolation time constant limit shape Eden model

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Auffinger, Antonio; Tang, Si. On the time constant of high dimensional first passage percolation. Electron. J. Probab. 21 (2016), paper no. 24, 23 pp. doi:10.1214/16-EJP1. https://projecteuclid.org/euclid.ejp/1459880112