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Rothschild Distinguished Visiting Fellow Lecture: Random maps and random 2-dimensional geometries

Monday 2nd February 2015 - 16:00 to 17:00
INI Seminar Room 1
A map is a graph embedded into a 2-dimensional surface, considered up to homeomorphisms. In a way, such an object endows the surface with a discrete metric, so that a map taken at random is a natural candidate for a notion of "discrete random metric surface". More precisely, it is expected (and proved in an evergrowing number of cases) that upon re-scaling the distances in an appropriate fashion, a large random map converges to a random metric space that is homeomorphic to the surface one started with.

This is reminiscent of the well-known convergence of random walks to Brownian motion. Similarly to the latter, the random continuum objects that appear as scaling limits of random maps are very irregular spaces, by no means close to being smooth Riemannian manifolds. This makes their study even more interinsting, since it is necessary to find the geometric notions that still make sense in this context, like geodesic paths.

By contrast with these "continuous" notions, we will see that the study of scaling limits of random maps relies strongly on tools of a purely combinatorial nature. We will also discuss conjectures which, quite surprisingly, connect these scaling limits with conformally invariant random fields in the plane.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons