Asymptotic normality of pattern occurrences in random maps

Abstract

The purpose of this paper is to study the limiting distribution of special additive functionals on random planar maps, namely the number of occurrences of a given pattern. The main result is a central limit theorem for these pattern counts in the case of patterns 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 [Probab. Theory Relat. Fields 130 (2004), 368–376]. It is an important issue to handle the overlap structure of two patterns which is the main difficulty in the proof.