Nesting statistics in the O(n) loop model on random maps of any topology

par Elba Garcia-Failde (MPIM, Bonn)

vendredi 6 janv. 2017, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Institut Camille Jordan)

In this talk, I will call maps a certain class of graphs embedded on surfaces. We will consider random maps equipped with a statistical physics model: the O(n) loop model. We investigate the nesting properties of loops by associating to every map with a loop configuration a so-called nesting graph, which encodes all the interesting nesting information. We study the generating series of maps of genus g with k boundaries and k' marked points realizing a fixed nesting graph, which are amenable to explicit computations in the loop model with bending energy on triangulations. We describe their critical behavior in the dense and dilute phases, arriving to some interesting qualitative conclusions about the most probable kind of maps. This was first studied for maps with the topology of disks and cylinders, and we generalize it to arbitrary topologies making use of a procedure called topological recursion, which will be introduced and applied to our analysis.