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Continuity of the time and isoperimetric constants in supercritical percolation. (English) Zbl 1388.60159
Summary: We consider two different objects on supercritical Bernoulli percolation on the edges of \(\mathbb{Z}^d\) : the time constant for i.i.d. first-passage percolation (for \(d\geq 2\)) 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 obtain 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 \(\mathbb{P} [t(e)<+\infty] >p_c(d)\). We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property of the asymptotic shape previously proved in [the second author, Adv. Appl. Probab. 12, 864–879 (1980; Zbl 0442.60096); the second author and H. Kesten, J. Appl. Probab. 18, 809–819 (1981; Zbl 0474.60085); H. Kesten, Lect. Notes Math. 1180, 125–264 (1986; Zbl 0602.60098)] for first-passage percolation with finite passage times.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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