The nested loop approach to the O(n) model on random maps

There is currently no consensus on what is the fractal (Hausdorff) 
dimension of a discrete random surface coupled to a critical matter 
model. In this talk, we describe a preliminary step in addressing this 
question, by considering the O(n) loop model on random planar maps (i.e. 
graphs embedded in the sphere). We explain how an elementary 
combinatorial decomposition, which consists in cutting the maps along 
the outermost loops, allows to relate the O(n) model to the simpler 
problem of counting maps with controlled face degrees (which may be 
solved using the classical Hermitian one-matrix model). This translates 
into a functional relation for the ``resolvent'' of the model, which is 
exactly solvable in several interesting cases. We then look for critical 
points of the model: our construction shows that at the so-called 
non-generic critical points, the O(n) model is related to the ``stable'' 
map, of known Hausdorff dimension, introduced by Le Gall and Miermont.

Based on joint work with G. Borot and E. Guitter.