December 2022 First-order behavior of the time constant in Bernoulli first-passage percolation
Anne-Laure Basdevant, Jean-Baptiste Gouéré, Marie Théret
Author Affiliations +
Ann. Appl. Probab. 32(6): 4535-4567 (December 2022). DOI: 10.1214/22-AAP1795


We consider the standard model of first-passage percolation on Zd (d2), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter 1ε. These passage times induce a random pseudo-metric Tε on Rd. By subadditive arguments, it is well known that for any zRd{0}, the sequence Tε(0,nz)/n converges a.s. toward a constant με(z) called the time constant. We investigate the behavior of εμε(z) near 0, and prove that με(z)=z1C(z)ε1/d1(z)+o(ε1/d1(z)), where d1(z) is the number of nonnull coordinates of z, and C(z) is a constant whose dependence on z is partially explicit.

Funding Statement

Research was partially supported by the ANR project PPPP (ANR-16-CE40-0016) and the Labex MME-DII (ANR 11-LBX-0023-01).


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Anne-Laure Basdevant. Jean-Baptiste Gouéré. Marie Théret. "First-order behavior of the time constant in Bernoulli first-passage percolation." Ann. Appl. Probab. 32 (6) 4535 - 4567, December 2022.


Received: 1 June 2021; Revised: 1 January 2022; Published: December 2022
First available in Project Euclid: 6 December 2022

MathSciNet: MR4522359
zbMATH: 07634774
Digital Object Identifier: 10.1214/22-AAP1795

Primary: 60K35 , 82B43

Keywords: Bernoulli random variables , First-passage percolation

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 6 • December 2022
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