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 .
Ann. Probab.
50(5):
1781-1812
(September 2022).
DOI: 10.1214/22-AOP1565
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