Volume 25, 2021
|Page(s)||109 - 132|
|Published online||23 March 2021|
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),
** Corresponding author: email@example.com
Accepted: 26 January 2021
We consider an i.i.d. supercritical bond percolation on ℤd, every edge is open with a probability p > pc(d), where pc(d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster 𝒞p. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ 𝒞p corresponds to the length of the shortest path in 𝒞p joining the two points. The chemical distance between 0 and nx grows asymptotically like nμp(x). We aim to study the regularity properties of the map p → μp in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is Gp = pδ1 + (1 − p)δ∞, p > pc(d). It is already known that the map p → μp is continuous.
Mathematics Subject Classification: 60K35 / 82B43
Key words: Regularity / percolation / time constant
© The authors. Published by EDP Sciences, SMAI 2021
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