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

Abstract

In the O(n) loop model on random planar maps, we study the depthin 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 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. 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, Watson and Wilson [111], or to its natural generalization 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.

Percolation probability and critical exponents for site percolation on the UIPT

“…We believe that it should be possible to adapt the strategy of this paper to other models of random planar maps and other statistical models. Indeed, what is needed first is the gasket decomposition; which exists for example for percolation on other models of maps [19], or for the O(n) model on maps [10,11]. We then need information on the generalized generating series of maps with a boundary and their singularities.…”

“…The second approach is more global and consists on decomposing the map into the cluster of the root vertex and pieces filling the faces of this cluster. Such a decomposition is often called the Gasket decomposition, see for instance the works of Borot, Bouttier, Duplantier and Guitter [10,11,12]. This second approach has been used very recently to study percolation on random finite triangulations by Bernardi, Curien and Miermont [7], following the previous work by Curien and Kortchemski [18]; and on other natural models of random finite planar maps by Curien and Richier [19].…”

The exploration process of critical Boltzmann planar maps decorated by a triangular $O(n)$ loop model

“…The techniques in [BBG12] and [BBD18] are, however, not quite sufficient to rigorously establish the phase diagram. This issue was resolved in the Appendix of [Bud18] for the rigid O(n) loop model.…”

“…This issue was resolved in the Appendix of [Bud18] for the rigid O(n) loop model. Nevertheless, the numerical derivation of the phase diagram in the physics papers [BBG12,BBD18] makes the following assumption plausible:…”

“…To be precise, it was shown in [BBD18] that given a fixed weight sequence q, the values γ + , γ − are uniquely determined and that in turn when γ + , γ − are fixed, the equation satisfied by the resolvent has a unique solution, which has to correspond to the partition functions of the loop-decorated random planar map (RPM) model. However, since in our situation q given by (2.3) already depends on the partition functions of the loopdecorated RPM model, it is not clear whether the aforementioned results of [BBD18] uniquely determine both the resolvent (2.4) and γ + , γ − provided that only (q, g, n) is known. Therefore, to prove the assumption it has to be shown that given only (q, g, n) the system of the two results from [BBD18] has a unique solution, which automatically has to be the resolvent (2.4) and γ + , γ − from the discussion above the Assumption.…”