The Continuous Limit of Large Random Planar Maps; A Renormalisation Group Analysis of the 4-Dimensional Continuous-time Weakly Self-avoiding Walk

1:35 – 2:00 Open problems (overlaps with lunch)
2:05 – 2:55 Jean-François Le Gall (Orsay)
The continuous limit of large random planar maps
3:05 – 3:45

Gordon Slade (U British Columbia)
A renormalisation group analysis of the
4-dimensional continuous-time weakly self-avoiding walk

The continuous limit of large random planar maps
Jean-François Le Gall (Université Paris-Sud, Orsay and Institut Universitaire de France).
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^-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, which is called the Brownian map, can be written as a quotient space of Aldous’ Continuum Random Tree (the CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. We discuss various properties of the Brownian map.

A renormalisation group analysis of the 4-dimensional continuous-time weakly self-avoiding walk
Gordon Slade (U British Columbia)
Abstract: We discuss recent joint work with David Brydges which proves |x|^-2 decay of the critical two-point function for the continuous-time weakly self-avoiding walk on Z⁴. The walk two-point function is identified as the two-point function of a supersymmetric field theory with quartic self-interaction, and the field theory is then analysed using renormalisation group methods.