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.