Modul: MAT971 Stochastische Prozesse

## Spine representations of non-compact models and isoperimetric inequalities in Brownian geometry

Talk by Dr. Armand Riera

**Date:** 03.11.21 **Time:** 17.15 - 18.15 **Room:** ETH HG G 19.1

We will provide a unified construction of the three main models of non-compact Brownian geometry, namely the Brownian plane, the infinite volume Brownian disk and the Brownian half-plane. We will explain how these three models can be encoded by a random infinite tree with non-negative labels under different - degenerated - conditionings. This new construction allows to understand the precise form of geodesics towards the boundary, to obtain a multiplicity of explicit computations and to investigate relations between these models. This part of the talk is based on a joint work with Jean-François Le Gall. We will then use this construction to establish sharp bounds on the probability of having a short cycle separating a large ball - centered in the root - from infinity in the Brownian plane. Finally, if time allows, we will combine all these results to determine the isoperimetric profil of the Brownian plane.