SEPARATING CYCLES AND ISOPERIMETRIC INEQUALITIES IN THE UNIFORM INFINITE PLANAR QUADRANGULATION

• 成果类型: Article
• 署名作者: Le Gall, Jean-Francois; Lehericy, Thomas
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1289
• 发表日期: 2019
• 页码: 1498-1540
• 关键词: growth

摘要

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 vertical bar A vertical bar(1/4)(log vertical bar A vertical bar)(-(3/4)-delta), where the volume vertical bar A vertical bar is the number of faces in A.