## The Annals of Probability

### Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation

#### 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 $A$.

#### Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1498-1540.

Dates
First available in Project Euclid: 2 May 2019

https://projecteuclid.org/euclid.aop/1556784025

Digital Object Identifier
doi:10.1214/18-AOP1289

Mathematical Reviews number (MathSciNet)
MR3945752

Zentralblatt MATH identifier
07067275

#### Citation

Le Gall, Jean-François; Lehéricy, Thomas. Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation. Ann. Probab. 47 (2019), no. 3, 1498--1540. doi:10.1214/18-AOP1289. https://projecteuclid.org/euclid.aop/1556784025

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