Abstract
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.
Highlights
The model of first passage percolation was first introduced by Hammersley and Welsh [9] as a model for the spread of a fluid in a porous medium
In [16], Cox and Kesten prove the continuity of this map in dimension 2 without any integrability condition. Their idea was to consider a geodesic for truncated passage times min(t(e), M ), and along it to avoid clusters of p-closed edges, that is to say edges with a passage time larger than some M > 0, by bypassing them with a short path in the boundary of this cluster
The edges of the boundary have passage time smaller than M. They were able to obtain a precise control on the length of these bypasses. This idea was later extended to all the dimensions d ≥ 2 by Kesten in [11], by taking a M large enough such that the percolation of the edges with a passage time larger than M is highly subcritical: for such a M, the size of the clusters of p-closed edges can be controlled
Summary
The model of first passage percolation was first introduced by Hammersley and Welsh [9] as a model for the spread of a fluid in a porous medium. In [16], Cox and Kesten prove the continuity of this map in dimension 2 without any integrability condition Their idea was to consider a geodesic for truncated passage times min(t(e), M ), and along it to avoid clusters of p-closed edges, that is to say edges with a passage time larger than some M > 0, by bypassing them with a short path in the boundary of this cluster. They were able to obtain a precise control on the length of these bypasses This idea was later extended to all the dimensions d ≥ 2 by Kesten in [11], by taking a M large enough such that the percolation of the edges with a passage time larger than M is highly subcritical: for such a M , the size of the clusters of p-closed edges can be controlled. The original part of this work is the quantification of the renormalization and the combinatorial estimates of Section 5
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