Electronic Journal of Probability

Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two

Julian Gold

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We study the isoperimetric subgraphs of the giant component $\mathbf{{C}} _n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\mathbf{{C}} _n$ with minimal edge boundary to volume ratio. In contrast to the work of [8], the edge boundary is taken only within $\mathbf{{C}} _n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of $\mathbf{{C}} _n$ scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 53, 41 pp.

Received: 14 May 2017
Accepted: 15 May 2018
First available in Project Euclid: 1 June 2018

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 52B60: Isoperimetric problems for polytopes

percolation Cheeger constant isoperimetry

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Gold, Julian. Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two. Electron. J. Probab. 23 (2018), paper no. 53, 41 pp. doi:10.1214/18-EJP178. https://projecteuclid.org/euclid.ejp/1527818431

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