We consider the standard model of first-passage percolation on (), 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 . These passage times induce a random pseudo-metric on . By subadditive arguments, it is well known that for any , the sequence converges a.s. toward a constant called the time constant. We investigate the behavior of near 0, and prove that , where is the number of nonnull coordinates of z, and is a constant whose dependence on z is partially explicit.
Research was partially supported by the ANR project PPPP (ANR-16-CE40-0016) and the Labex MME-DII (ANR 11-LBX-0023-01).
"First-order behavior of the time constant in Bernoulli first-passage percolation." Ann. Appl. Probab. 32 (6) 4535 - 4567, December 2022. https://doi.org/10.1214/22-AAP1795