Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation

TL;DR

This paper investigates the geometric structure of the uniform infinite planar quadrangulation (UIPQ), proving a linear growth of separating cycles and deriving isoperimetric inequalities that relate boundary size to volume.

Contribution

It establishes the linear growth of minimal separating cycles in UIPQ and derives new isoperimetric bounds involving volume and boundary size.

Findings

Minimal separating cycle size grows linearly with radius R.

Boundary size of finite face unions is bounded below by a function of volume.

Provides rigorous proof of Krikun's conjecture on cycle growth.

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 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 $A$.