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Tertiles and the time constant

Part of: Special processes Limit theorems Equilibrium statistical mechanics

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: [email protected]
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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

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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