Asymptotic normality of pattern counts in random maps II

TL;DR

This paper presents a shorter proof of the asymptotic normality of pattern counts in random maps, extending the result to various boundary conditions and new map classes using advanced asymptotic analysis.

Contribution

It introduces a novel proof technique that avoids reduction to face counts and generalizes the asymptotic normality results to broader classes of maps.

Findings

Shorter proof of asymptotic normality for pattern counts

Extension to maps with arbitrary boundary conditions

Application to new classes of maps

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.