Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T15:25:10.924Z Has data issue: false hasContentIssue false
  • English
  • Français

Tertiles and the time constant

Part of: Equilibrium statistical mechanics Special processes Limit theorems

Published online by Cambridge University Press:  16 July 2020

Daniel Ahlberg Open the ORCID record for Daniel Ahlberg [Opens in a new window]
Daniel Ahlberg*
Affiliation:
Stockholm University
*
*Postal address: Department of Mathematics, Stockholm University, SE - 106 91 Stockholm, Sweden. Email address: daniel.ahlberg@math.su.se
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider planar first-passage percolation and show that the time constant can be bounded by multiples of the first and second tertiles of the weight distribution. As a consequence, we obtain a counter-example to a problem proposed by Alm and Deijfen (2015).

Keywords

First-passage percolationquantilestime constant

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60F15: Strong theorems 82B43: Percolation
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 407 - 408
Copyright
© Applied Probability Trust 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alm, S. E. and Deijfen, M. (2015). First passage percolation on $\mathbb{Z}^2$ : A simulation study. J. Stat. Phys. 161, 657–678.10.1007/s10955-015-1356-0CrossRefGoogle Scholar
Auffinger, A., Damron, M. and Hanson, J. (2017). 50 Years of First-Passage Percolation (Univ. Lect. Ser. 68). American Mathematical Society, Providence, RI.10.1090/ulect/068CrossRefGoogle Scholar
Cox, J. T. (1980). The time constant of first-passage percolation on the square lattice. Adv. Appl. Prob. 12, 864–879.10.2307/1426745CrossRefGoogle Scholar
Durrett, R. and Liggett, T. M. (1981). The shape of the limit set in Richardson’s growth model. Ann. Prob. 9, 186–193.10.1214/aop/1176994460CrossRefGoogle Scholar
Liggett, T. M. (1995). Survival of discrete time growth models, with applications to oriented percolation. Ann. Appl. Prob. 5, 613–636.10.1214/aoap/1177004698CrossRefGoogle Scholar
Smythe, R. T. and Wierman, J. C. (1978). First-Passage Percolation on the Square Lattice (Lect. Notes Math. 671). Springer, Berlin.10.1007/BFb0063306CrossRefGoogle Scholar