Compact and non-compact models of random geometry

Abstract: We discuss various models of random geometry that arise as scaling limits of large planar graphs embedded in the 2-sphere (also called planar maps). The most popular compact models are the Brownian sphere or Brownian map, and the Brownian disk, which is the scaling limit of planar maps with a boundary. We explain how Brownian disks can be viewed as connected components of the complement of balls in the Brownian sphere, and we discuss a remarkable growth-fragmentation process that describes the evolution of the boundary sizes of these components when the radius of the ball increases. We also introduce the non-compact models called the Brownian plane, the infinite Brownian disk and the Brownian half-plane, and we present a unified construction of these three models based on a spine decomposition. Most of the talk is based on joint work with Armand Riera.

mathematical physicsprobability

Audience: researchers in the topic

( slides )

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Probability and the City Seminar

Series comments: The Probability and the City Seminar is organized jointly by the probability groups of Columbia University and New York University.

Video recordings of talks are posted online at www.youtube.com/channel/UC0CXjG-ZSIZHy0S40Px2FEQ .

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