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

Understand

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.