🤖 AI Summary
This study investigates the asymptotic normality of pattern counts in random planar maps. By directly analyzing the bivariate coefficient asymptotics of functional equations involving a single catalytic variable, the authors establish a central limit theorem for pattern occurrences without relying on prior assumptions about face-degree distributions. The approach leverages techniques from analytic combinatorics—specifically, functional equations and bivariate asymptotic analysis—to yield a more streamlined proof while extending the result to arbitrary boundary conditions and broader classes of maps. This work substantially broadens the scope under which asymptotic normality holds, enhancing both the generality and technical efficiency of the underlying theory.
📝 Abstract
In a recent work, a central limit theorem for pattern counts in random planar maps was proven by reducing the problem to a face count problem. We provide a shorter proof by circumventing this reduction through the computation of bivariate coefficient asymptotics from a functional equation with one catalytic variable and extend the result to pattern counts with arbitrary boundary and new map classes.