Isoperimetric inequalities in the Brownian plane

TL;DR

This paper investigates the geometric properties of the Brownian plane, establishing bounds on short cycles and isoperimetric inequalities, and demonstrating a strong spatial Markov property for this universal scaling limit of random planar maps.

Contribution

It provides the first sharp bounds on separating cycles and proves a strong spatial Markov property for the Brownian plane, advancing understanding of its geometric structure.

Findings

Sharp bounds on the probability of short separating cycles.

A strong version of the spatial Markov property.

Sharp isoperimetric inequalities for the Brownian plane.

Abstract

We consider the model of the Brownian plane, which is a pointed non-compact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar triangulation or the uniform infinite planar quadrangulation and is conjectured to be the universal scaling limit of many others random planar lattices. We establish sharp bounds on the probability of having a short cycle separating the ball of radius $r$ centered at the distinguished point from infinity. Then we prove a strong version of the spatial Markov property of the Brownian plane. Combining our study of short cycles with this strong spatial Markov property we obtain sharp isoperimetric bounds for the Brownian plane.