We consider the standard model of i.i.d. first passage percolation on given a distribution G on ( is allowed). When , it is known that the time constant exists. We are interested in the regularity properties of the map . We first study the specific case of distributions of the form for . 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 is Lipschitz continuous on every interval , where .
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.
"The time constant for Bernoulli percolation is Lipschitz continuous strictly above ." Ann. Probab. 50 (5) 1781 - 1812, September 2022. https://doi.org/10.1214/22-AOP1565