September 2022 The time constant for Bernoulli percolation is Lipschitz continuous strictly above pc
Raphaël Cerf, Barbara Dembin
Author Affiliations +
Ann. Probab. 50(5): 1781-1812 (September 2022). DOI: 10.1214/22-AOP1565


We consider the standard model of i.i.d. first passage percolation on Zd given a distribution G on [0,+] (+ is allowed). When G([0,+))>pc(d), it is known that the time constant μG exists. We are interested in the regularity properties of the map GμG. We first study the specific case of distributions of the form Gp=pδ1+(1p)δ for p>pc(d). In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter p. We show that the function pμGp is Lipschitz continuous on every interval [p0,1], where p0>pc(d).

Funding Statement

The second author would like to thank Association Séphora Berrebi for their support. The second author was supported by SNF Grant 175505 and is part of NCCR SwissMAP.


Dedicated to the memory of Vladas Sidoravicius


The second author is laureate of the Séphora Berrebi Scholarship in Mathematics in 2019.


Download Citation

Raphaël Cerf. Barbara Dembin. "The time constant for Bernoulli percolation is Lipschitz continuous strictly above pc." Ann. Probab. 50 (5) 1781 - 1812, September 2022.


Received: 1 February 2021; Revised: 1 September 2021; Published: September 2022
First available in Project Euclid: 24 August 2022

MathSciNet: MR4474502
zbMATH: 07603474
Digital Object Identifier: 10.1214/22-AOP1565

Primary: 60K35
Secondary: 82B43

Keywords: first passage percolation , percolation , Time constant

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 5 • September 2022
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