We consider an i.i.d. supercritical bond percolation on , every edge is open with a probability , where denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster 𝒞$$. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y ∈ 𝒞$$ corresponds to the length of the shortest path in 𝒞$$ joining the two points. The chemical distance between 0 and nx grows asymptotically like nμ$$(x). We aim to study the regularity properties of the map p → μ$$ in the supercritical regime. This may be seen as a special case of first passage percolation where the distribution of the passage time is G$$ = pδ1 + (1 − p)δ$$, p > p$$(d). It is already known that the map p → μ$$ is continuous.
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DOI : 10.1051/ps/2021005
Keywords: Regularity, percolation, time constant
@article{PS_2021__25_1_109_0,
author = {Dembin, Barbara},
title = {Regularity of the time constant for a supercritical {Bernoulli} percolation},
journal = {ESAIM: Probability and Statistics},
pages = {109--132},
year = {2021},
publisher = {EDP-Sciences},
volume = {25},
doi = {10.1051/ps/2021005},
mrnumber = {4234130},
zbl = {1468.60116},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2021005/}
}
TY - JOUR AU - Dembin, Barbara TI - Regularity of the time constant for a supercritical Bernoulli percolation JO - ESAIM: Probability and Statistics PY - 2021 SP - 109 EP - 132 VL - 25 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2021005/ DO - 10.1051/ps/2021005 LA - en ID - PS_2021__25_1_109_0 ER -
Dembin, Barbara. Regularity of the time constant for a supercritical Bernoulli percolation. ESAIM: Probability and Statistics, Tome 25 (2021), pp. 109-132. doi: 10.1051/ps/2021005
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Cité par Sources :
Research was partially supported by the ANR project PPPP (ANR-16-CE40-0016). This study contributes to the IdEx Université de Paris ANR-18-IDEX-0001.
