A probabilistic approach to block sizes in random maps

TL;DR

This paper introduces a probabilistic method to analyze block sizes in random maps, providing simpler proofs and new distributional convergence results for the sizes of large 2-connected blocks.

Contribution

It offers a novel probabilistic approach that simplifies existing proofs and establishes new distributional convergence results for block sizes in large random maps.

Findings

Convergence in distribution of rescaled sizes of the k'th largest 2-connected block to a Fréchet distribution.

Simplified proofs of previous results without singularity analysis.

New results for the case when k=2 in block size distribution.

Abstract

We present a probabilistic approach to the core-size in random maps, which yields straightforward and singularity analysis-free proofs of some results of Banderier, Flajolet, Schaeffer and Soria. The proof also yields convergence in distribution of the rescaled size of the k'th largest 2-connected block in a large random map, for any fixed k > 1, to a Fr\'echet-type extreme order statistic. This seems to be a new result even when k=2.