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2018 Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two
Julian Gold
Electron. J. Probab. 23: 1-41 (2018). DOI: 10.1214/18-EJP178

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.

Citation

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Julian Gold. "Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two." Electron. J. Probab. 23 1 - 41, 2018. https://doi.org/10.1214/18-EJP178

Information

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

zbMATH: 06924665
MathSciNet: MR3814247
Digital Object Identifier: 10.1214/18-EJP178

Subjects:
Primary: 52B60, 60K35, 82B43

JOURNAL ARTICLE
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