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

Abstract

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.

Dedication

Dedicated to the memory of Vladas Sidoravicius

Acknowledgments

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

Citation

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

Information

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

Digital Object Identifier: 10.1214/22-AOP1565

Subjects:
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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