Nesting statistics in the $O(n)$ loop model on random planar maps

In the O(n) loop model on random planar maps, we study the depth – in terms of the number of levels of nesting – of the loop configuration, by means of analytic combinatorics. We focus on the " refined " generating series of pointed disks or cylinders, which keep track of the number of loops separating the marked point from the boundary (for disks), or the two boundaries (for cylinders). For the general O(n) loop model, we show that these generating series satisfy functional relations obtained by a modification of those satisfied by the unrefined generating series. In a more specific O(n) model where loops cross only triangles and have a bending energy, we can explicitly compute the refined generating series. We analyze their non-generic critical behavior in the dense and dilute phases, and obtain the large deviations function of the nesting distribution, which is expected to be universal. By a continuous generalization of the KPZ relation in Liouville quantum gravity, i.e., by taking into account the probability distribution of the Euclidean radius of a ball of given quantum area, our results are in full agreement with the multifractal spectra of extreme nesting of CLE κ in the disk, as obtained by Miller, Watson and Wilson [71], and with its natural generalization to the Riemann sphere