Gaskets of $O(2)$ loop-decorated random planar maps
Emmanuel Kammerer
TL;DR
This work establishes that at the critical point $n=2$, the gaskets of rigid $O(2)$ loop-decorated random planar maps are $3/2$-stable maps, with gasket partition functions $W^{(\ell)}$ exhibiting $W^{(\ell)} \sim c_{\bf q}^{\ell+1} L_{\bf q}(\ell)/(2\ell^{2})$, where $L_{\bf q}$ is a slowly varying function (often of order $\log\ell$). The authors achieve this via a Wiener–Hopf factorisation framework for a derived random walk $S$, using the pre-renewal function $h^\downarrow$ and a carefully constructed measure $\nu$ to connect gasket weights to ladder-height processes; this yields a dichotomy in the tail of $\nu(-k)$ depending on $\sum_{j\ge1} j g_j$. In addition to proving the main gasket asymptotics, the paper characterises weight sequences $\bf q$ corresponding to critical $O(2)$ decorations and provides explicit examples (including Budd’s symmetric case and fully packed quadrangulations) that illustrate how slowly varying factors arise in this critical setting. Overall, the results strengthen evidence that critical $O(2)$ loop-decorated maps converge to a critical Liouville quantum gravity disk decorated by CLE$_4$, and they illuminate the interplay between random-walk harmonicity, Wiener–Hopf factorisation, and the geometry of gasket-driven scaling limits.
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.