Title: The O(n) loop model on random planar maps
The O(n) model can be formulated in terms of loops living on the lattice,
with n the fugacity per loop. In two dimensions, it is known to possess
a rich critical behavior, involving critical exponents varying continuously
with n. In this talk, we will consider the case where the model is "coupled
to 2D quantum gravity", namely it is defined on a fluctuating 2D random
lattice (i.e. a random map).
It has been known since the 90's that the partition function of the model
can be expressed as a matrix integral, which can be evaluated exactly in
the planar limit. More recently, we have revisited the problem by a
combinatorial approach, which allows to relate it to a more elementary
model of maps without loops. In particular we establish that the critical
points of the O(n) model are closely related to the "maps with large faces"
considered by Le Gall and Miermont. Finally, our approach allows to study
the statistics of nestings between loops (how many loops wind around a
randomly chosen point, etc).
This talk is based on joint works (some of which are in progress) with
Gaetan Borot, Emmanuel Guitter and Bertrand Duplantier.