Regularity of the time constant for a supercritical Bernoulli percolation^*

Université de Paris, Sorbonne Université, CNRS, Laboratoire de Probabilités, Statistique et Modélisation (LPSM, UMR 8001), 75006 Paris, France.

^** Corresponding author: bdembin@lpsm.paris

Received: 7 January 2019
Accepted: 26 January 2021

Abstract

We consider an i.i.d. supercritical bond percolation on ℤ^d, every edge is open with a probability p > p_c(d), where p_c(d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster 𝒞_p. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ 𝒞_p corresponds to the length of the shortest path in 𝒞_p joining the two points. The chemical distance between 0 and nx grows asymptotically like nμ_p(x). We aim to study the regularity properties of the map p → μ_p in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is G_p = pδ₁ + (1 − p)δ_∞, p > p_c(d). It is already known that the map p → μ_p is continuous.