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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.

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