More on the O(n) model on random maps via nested loops: loops with bending energy

Understand

We continue our investigation of the nested loop approach to the O(n) model on random maps, by extending it to the case where loops may visit faces of arbitrary degree.

• This allows to express the partition function of the O(n) loop model as a specialization of the multivariate generating function of maps with controlled face degrees, where the face weights are determined by a fixed point condition.
• We deduce a functional equation for the resolvent of the model, involving some ring generating function describing the immediate vicinity of the loops.
• When the ring generating function has a single pole, the model is amenable to a full solution.