Unified study of the phase transition for block-weighted random planar maps

TL;DR

This paper extends a methodology to analyze phase transitions in block-weighted random maps, demonstrating their existence and detailing the largest block sizes across different regimes.

Contribution

It generalizes previous methods to a broader class of maps, establishing the presence of phase transitions and characterizing block sizes.

Findings

Phase transition exists in various map families.

Largest block sizes vary across regimes.

Methodology is broadly applicable.

Abstract

In [Fleurat, Salvy 2024], we introduced a model of block-weighted random maps that undergoes a phase transition as the density of separating elements changes. The purpose of this note is to demonstrate that the methodology we developed can be extended to many other families of maps. We prove that a phase transition exists and provide detailed information about the size of the largest blocks in each regime.