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

We prove that for $n=2$ the gaskets of critical rigid $O(n)$ loop-decorated random planar maps are $3\mathrm{/}2$-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.