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Separating cycles and isoperimetric inequalities in the uniform infinite planar quadrangulation. (English) Zbl 1414.05267

Summary: 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 M. Krikun [“Local structure of random quadrangulations”, Preprint, arXiv:math/0512304] 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 \(A\).

MSC:

05C80 Random graphs (graph-theoretic aspects)
60D05 Geometric probability and stochastic geometry
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References:

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