Electronic Journal of Probability

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

Julian Gold

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1527818431

Digital Object Identifier
doi:10.1214/18-EJP178

Mathematical Reviews number (MathSciNet)
MR3814247

Zentralblatt MATH identifier
06924665

Subjects
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

Keywords
percolation Cheeger constant isoperimetry

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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