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

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.

arXiv comment20 pages, 2 figures. To appear in Mathematical Physics, Analysis and Geometry