We consider the model of the Brownian plane, which is a pointed noncompact 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.
The present work was supported by the ERC Advanced Grant 740943 GEOBROWN.
I warmly thank Jean-François Le Gall for all his suggestions and for carefully reading earlier versions of this manuscript. We also thank the referee for several useful comments.
"Isoperimetric inequalities in the Brownian plane." Ann. Probab. 50 (5) 2013 - 2055, September 2022. https://doi.org/10.1214/22-AOP1576