Sigurður Örn Stefánsson (27/10/14)

Benedikt Magnússon, October 23, 2014

Math Colloquium

Speaker: Sigurður Örn Stefánsson, University of Iceland
Title: Convergence of random planar maps to the Brownian tree

Location: V-157, VRII
Time: Monday, October 27 at 15:15-16:15.

Abstract:

Random planar maps are defined by assigning non-negative weights to each face of a planar map and the weight of a face depends only on its degree. I will explain the Bouttier-Di Francesco-Guitter bijection between the planar maps and a class of labelled trees called mobiles. By throwing away labels one can, via another bijection, relate the mobiles to the model of so-called simply generated trees which are understood in detail. For certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears with high probability when the maps are large. This corresponds to a recently studied phenomenon of condensation in simply generated trees where a vertex having degree proportional to the size of the trees appears. In this case the planar maps, with a properly rescaled graph metric, are shown to converge in distribution towards Aldous’ Brownian tree in the Gromov-Hausdorff topology.