Monna Lecture
Abstract: Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they also have significant geometrical applications. Random planar maps have been used in theoretical physics, where they serve as models of random geometry. Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all planar maps with n vertices in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power -1/4. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces.
This problem was stated by Oded Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit can be written as a quotient space of the so-called Continuum Random Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels atttached to the vertices of the CRT. This limiting random metric space, which is called the Brownian map, can be viewed as a "Brownian surface" in the same sense as Brownian motion is the limit of rescaled discrete paths. We show that the Brownian map is almost surely homeomorphic to the two-dimensional sphere, although it has Hausdorff dimension 4. As a key tool, we use bijections between planar maps and various classes of labeled trees.
Here is a link to the slides of Le Gall’s talk.
Attention! The Monna Lecture will take place in MIN 211
Location: Minnaert 211 of the Wiskunde building (campus De Uithof) Budapestlaan 6, Utrecht.
Date and time: Thursday, September 24, 2009 15:30-16:30. The lecture will be followed by a small drinks reception in the same room.
Schedule of the Utrecht math colloquium : Next term’s talks, This year's talks. Last year's talks. Guidelines for speakers.
Other math colloquia : TU Delft, Leiden, UvA Amsterdam, VU Amsterdam, Nijmegen.
Organisers : Gil Cavalcanti, Wilberd van der Kallen and Paul Zegeling