Gaskets of $O(2)$ loop-decorated random planar maps

TL;DR

This paper proves that gaskets of critical $O(2)$ loop-decorated random planar maps are $3/2$-stable maps, using Wiener-Hopf factorisation, and characterizes their weight sequences.

Contribution

It establishes the critical case for $n=2$ in $O(n)$ loop-decorated maps as $3/2$-stable maps and provides a new characterization of their weight sequences.

Findings

Gaskets of critical $O(2)$ maps are $3/2$-stable maps.

The proof uses Wiener-Hopf factorisation for random walks.

Provides a characterization of critical $O(2)$ map weight sequences.

Abstract

We prove that for $n = 2$ the gaskets of critical rigid $O(n)$ loop-decorated random planar maps are $3/2$-stable maps. The case $n = 2$ thus corresponds to the critical case in random planar maps. The proof relies on the Wiener-Hopf factorisation for random walks. Our techniques also provide a characterisation of weight sequences of critical $O(2)$ loop-decorated maps.