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 explicitly compute the refined generating series. We analyse 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. Using the framework of Liouville quantum gravity (LQG), we show that a rigorous functional KPZ relation can be applied to the multifractal spectrum of extreme nesting in the conformal loop ensemble ( CLE_κ ) in the Euclidean unit disk, as obtained by Miller et al. (Ann Probab 44(2):1013–1052, 2016, arXiv:1401.0217 ), or to its natural generalisation to the Riemann sphere. It allows us to recover the large deviations results obtained for the critical O ( n ) random planar map models. This offers, at the refined level of large deviations theory, a rigorous check of the fundamental fact that the universal scaling limits of random planar map models as weighted by partition functions of critical statistical models are given by LQG random surfaces decorated by independent CLEs.