Institute of Mathematics

Talk

Modul:   MAT971  Stochastische Prozesse

On scaling limits of random planar maps

Prof. Dr. Cyril Marzouk talk

Speaker invited by: Prof. Dr. Jean Bertoin

Date: 25.03.20  Time: 17.15 - 19.00  Room: Y27H12 CANCELLED

A planar map is a surface homeomorphic to a 2-sphere obtained by (topologically) gluing polygons together. In 2011 Grégory Miermont proved that a uniform random gluing of quadrangles converges in distribution as the number of quadrangles tends to infinity, once properly rescaled, to a continuum random object called the Brownian map, in the same spirit that the rescaled simple random walk converges to the Brownian motion; this result was simultaneous obtained by Jean-François Le Gall who also established the convergence to the same limit of uniform random gluing of triangles, or of any fixed polygon (with even number of sides…). After recalling the basics of this theory, I will discuss some recent extension of this result, in the spirit of Donsker’s Theorem, when more generally one takes a uniform random gluing of an arbitrary list of polygons: the Brownian map arises in the limit as soon as there is no "macroscopic" polygon.