Asymptotic normality of pattern occurrences in random maps

TL;DR

This paper proves a central limit theorem for the number of pattern occurrences in random planar maps with simple boundaries, using analytic, combinatorial, and moment methods to handle overlaps.

Contribution

It introduces a new CLT for pattern counts in random maps, addressing overlap complexities with a novel combination of methods.

Findings

Pattern counts follow a normal distribution asymptotically.

The proof combines analytic, combinatorial, and moment methods.

Handles overlap structures in pattern occurrences effectively.

Abstract

The purpose of this paper is to study the limiting distribution of special {\it additive functionals} on random planar maps, namely the number of occurrences of a given {\it pattern}. The main result is a central limit theorem for these pattern counts in the case of pattern with a simple boundary. The proof relies on a combination of analytic and combinatorial methods together with a moment method due to Gao and Wormald~\cite{GaoWormald}. It is an important issue to handle the overlap structure of two pattern which is the main difficulty in the proof.