SEPARATING CYCLES AND ISOPERIMETRIC INEQUALITIES IN THE UNIFORM INFINITE PLANAR QUADRANGULATION

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...

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Bibliographic Details
Published in:The Annals of probability 2019-05, Vol.47 (3), p.1498-1540
Main Authors: Le Gall, Jean-François, Lehéricy, Thomas
Format: Article
Language:English
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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 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)−δ , where the volume |A| is the number of faces in A.
ISSN:0091-1798
2168-894X
DOI:10.1214/18-AOP1289