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 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 $A$.

Revised version, 47 pages, to appear in the Annals of Probability