Abstract
We consider two different objects on super-critical Bernoulli percolation on $\mathbb{Z}^d$ : the time constant for i.i.d. first-passage percolation (for $d\geq2$) and the isoperimetric constant (for $d=2$). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in $\mathbb{Z}^2$ is continuous in the percolation parameter. As a corollary we prove that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdorff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on $\mathbb{Z}^d$ with possibly infinite passage times: we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbb{P}[t(e) p_c(d)$. We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property previously proved by Cox and Kesten for first passage percolation with finite passage times.
Highlights
We consider supercritical bond percolation on Zd, with parameter p > pc(d), the critical parameter for this percolation
We study the continuity properties of two distinct objects defined on this infinite cluster: the isoperimetric constant, and the asymptotic shape for an independent first-passage percolation
Notice that if Gn →d G and G([0, +∞)) > pc(d) Gn([0, +∞)) > pc(d) at least for n large enough. This result extends the continuity of the time constant in classical first-passage percolation proved by Cox and Kesten [8, 10, 20] to first-passage percolation with possibly infinite passage times
Summary
We consider supercritical bond percolation on Zd, with parameter p > pc(d), the critical parameter for this percolation. We study the continuity properties of two distinct objects defined on this infinite cluster: the isoperimetric (or Cheeger) constant, and the asymptotic shape (or time constant) for an independent first-passage percolation. Notice that if Gn →d G and G([0, +∞)) > pc(d) Gn([0, +∞)) > pc(d) at least for n large enough This result extends the continuity of the time constant in classical first-passage percolation proved by Cox and Kesten [8, 10, 20] to first-passage percolation with possibly infinite passage times. P ∈ (pc(d), 1] → Bp is continuous for the Hausdorff distance between non-empty compact sets of Rd. As a key step of the proof of Theorem 1.2, we study the effect of truncations of the passage time on the time constant.
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