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

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 us 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. Physically, such a situation is realized upon considering loops visiting triangles only and further weighting these loops by some local bending energy. Our model interpolates between the two previously solved cases of triangulations without bending energy and quadrangulations with rigid loops. We analyze the phase diagram of our model in details and derive in particular the location of its non-generic critical points, which are in the universality classes of the dense and dilute O(n) model coupled to 2D quantum gravity. Similar techniques are also used to solve a twisting loop model on quadrangulations where loops are forced to make turns within each visited square. Along the way, we revisit the problem of maps with controlled, possibly unbounded, face degrees and give combinatorial derivations of the one-cut lemma and of the functional equation for the resolvent.