Journée "Cartes Aléatoires"

Hyperbolic random maps : an overview

by Prof. Gourab RAY (Université de Cambridge)

Centre de conférences Marilyn et James Simons (I.H.E.S.)

Centre de conférences Marilyn et James Simons


Le Bois-Marie 35, route de Chartres 91440 Bures-sur-Yvette
Recently, hyperbolic versions of uniform planar maps have attracted a great deal of attention. These maps are conjectured to be local limits of uniform maps embedded on high genus surfaces. First, I will describe a resolution of this conjecture for unicellular (or one-face) maps. Although for other cases this still remains a conjecture, several possible candidates have been constructed. I will give a brief overview of these models, their construction and geometric properties. I will also discuss behaviour of random walks (e.g. their speed) on them and how the ''final behaviour" of random walks on them can be nicely described via their circle packings. Parts of these works are joint with Omer Angel, Guillaume Chapuy, Nicolas Curien, Tom Hutchcroft and Asaf Nachmias.