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May 2019 Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation
Jean-François Le Gall, Thomas Lehéricy
Ann. Probab. 47(3): 1498-1540 (May 2019). DOI: 10.1214/18-AOP1289

Abstract

We study geometric properties of the infinite random lattice called the uniform infinite planar quadrangulation or UIPQ. We establish a precise form of a conjecture of Krikun stating that the minimal size of a cycle that separates the ball of radius $R$ centered at the root vertex from infinity grows linearly in $R$. As a consequence, we derive certain isoperimetric bounds showing that the boundary size of any simply connected set $A$ consisting of a finite union of faces of the UIPQ and containing the root vertex is bounded below by a (random) constant times $|A|^{1/4}(\log|A|)^{-(3/4)-\delta}$, where the volume $|A|$ is the number of faces in $A$.

Citation

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Jean-François Le Gall. Thomas Lehéricy. "Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation." Ann. Probab. 47 (3) 1498 - 1540, May 2019. https://doi.org/10.1214/18-AOP1289

Information

Received: 1 October 2017; Published: May 2019
First available in Project Euclid: 2 May 2019

zbMATH: 07067275
MathSciNet: MR3945752
Digital Object Identifier: 10.1214/18-AOP1289

Subjects:
Primary: 05C80
Secondary: 60D05

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 3 • May 2019
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