Title: 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)-δ , where the volume |A| is the number of faces in A.