Electronic Communications in Probability

Vanishing of the anchored isoperimetric profile in bond percolation at $p_{c}$

Raphaël Cerf and Barbara Dembin

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We consider the anchored isoperimetric profile of the infinite open cluster, defined for $p>p_{c}$, whose existence has been recently proved in [3]. We extend adequately the definition for $p=p_{c}$, in finite boxes. We prove a partial result which implies that, if the limit defining the anchored isoperimetric profile at $p_{c}$ exists, it has to vanish.

Article information

Electron. Commun. Probab., Volume 25 (2020), paper no. 2, 7 pp.

Received: 18 March 2019
Accepted: 8 December 2019
First available in Project Euclid: 4 January 2020

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

critical percolation isoperimetric constant

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Cerf, Raphaël; Dembin, Barbara. Vanishing of the anchored isoperimetric profile in bond percolation at $p_{c}$. Electron. Commun. Probab. 25 (2020), paper no. 2, 7 pp. doi:10.1214/19-ECP281. https://projecteuclid.org/euclid.ecp/1578106940

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