Electronic Journal of Probability

On the time constant of high dimensional first passage percolation

Antonio Auffinger and Si Tang

Full-text: Open access

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

Digital Object Identifier
doi:10.1214/16-EJP1

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


Export citation

References

  • [1] Antonio Auffinger, Michael Damron, and Jack Hanson, 50 years of first passage percolation, arXiv.org (2015), 1–148.
  • [2] Olivier Couronné, Nathanaël Enriquez, and Lucas Gerin, Construction of a short path in high-dimensional first passage percolation, Electron. Commun. Probab. 16 (2011), 22–28.
  • [3] J. Theodore Cox and Richard Durrett, Oriented percolation in dimensions $d\geq 4$: bounds and asymptotic formulas, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 1, 151–162.
  • [4] Deepak Dhar, First passage percolation in many dimensions, Phys. Lett. A 130 (1988), no. 4–5, 308–310.
  • [5] John Michael Hammersley, Long-Chain Polymers and Self-Avoiding Random Walks, Sankhyā: The Indian Journal of Statistics, Series A 25 (1963), no. 1, 29–38.
  • [6] Harry Kesten, On the number of Self-Avoiding Walks. II, Journal of Mathematical Physics 5 (1964), no. 8, 1128–1137.
  • [7] Harry Kesten, Aspects of first passage percolation, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 125–264.
  • [8] Neal Madras and Gordon Slade, The Self-Avoiding Walk, Springer Science & Business Media, Boston, MA, November 2013.
  • [9] Anders Martinsson, First-passage percolation on cartesian power graphs, arXiv.org (2015), 1–27.
  • [10] Jacob van den Berg and Harry Kesten, Inequalities for the time constant in first-passage percolation, Ann. Appl. Probab. 3 (1993), no. 1, 56–80.