Grégory Miermont received his education at Ecole Normale Supérieure (ENS) in Paris from 1998–2002. He defended his PhD thesis, which was supervised by Jean Bertoin, in 2003. He also spent a year in Berkeley, 2001–02, working under the supervision of David Aldous and Jim Pitman. Miermont held a CNRS researcher position from 2004–09, first in Orsay and then in Paris (Université Pierre et Marie Curie and ENS). From 2009–2012 he was a professor at Université Paris-Sud (Orsay), on leave to the University of British Columbia during 2011-2012. Since 2012, he is a professor at ENS Lyon. His research, in the area of random trees and random planar maps, has been recognized by several distinctions, including the 2009 Rollo Davidson Prize, the 2012 EMS Prize and the 2014 Wolfgang Döblin Prize. This Medallion Lecture will be given at SPA 2015 in Oxford, UK, July 13–17, 2015 (where there will also be the Schramm lecture by Michel Ledoux and another Medallion lecture by Scott Sheffield).

Compact Brownian surfaces

In recent years, it has been shown how random maps can be viewed as a discrete version for a canonically defined “random surface”. Consider for instance a triangulation of the 2-dimensional sphere into $n$ triangles, chosen uniformly among the (finite) set of non-combinatorially equivalent such triangulations. The resulting random discrete sphere can be endowed with a natural distance function, for instance the graph distance on the set of vertices of the map.

One can try to renormalize these distances as the size $n$ goes to the infinity, in order to obtain a meaningful scaling limit for these random metric spaces, which would be a compact, non-trivial random metric space, the “Brownian map”. This has been the object of active research over the past 10 years, starting with a pioneering work by Chassaing and Schaeffer, who were able in particular to identify the appropriate renormalization rate to be $n^{1/4}$. The last step of the proof of convergence, consisting in uniquely identifying the limiting Brownian map, has been established in 2011 in two independent works by Miermont (dealing in fact the case of quadrangulations of the sphere) and Le Gall (for more general models of random maps).

The work I am going to discuss is a joint work with Jérémie Bettinelli, in which we establish a similar convergence result for random maps defined on orientable compact surfaces of the most general topology, and possibly with a boundary. This is achieved by using appropriate surgical operations, starting for appropriately conditioned version of the Brownian map, or its sibling, the Brownian disk. The latter is in turn defined as a gluing of infinitely many “slices”, obtained by cutting the Brownian map along a marked geodesic path.