Universal limiting shape for rescaled Dyck paths

Determine whether the rescaled piecewise linear functions associated with monotonically ordered non-crossing pair-partitions converge, under averaging over \(N C^{(mton)}_2(2n)\) as \(n\to\infty\), to a universal limiting function \(g:[0,2]\to[0,\infty)\).

Background

The paper studies the area statistic of monotonically ordered non-crossing pair-partitions and proves that its expectation grows asymptotically like $n\ln n$. Motivated by this growth rate, the authors rescale the associated piecewise linear Dyck-path functions $f_\pi$ by defining $g_\pi(s)=f_\pi(ns)/\ln n$ on $[0,2]$.

Theorem 1.5 implies that the expected area of the rescaled functions converges to 1. The unresolved question is whether this convergence of the integrated statistic reflects convergence, after averaging over the pair-partition space, of the functions themselves to a universal deterministic profile.