The scaling limit of the volume of loop O(n) quadrangulations

TL;DR

The authors determine the scaling limit for the volume $V$ of rigid loop-$O(n)$ quadrangulations with boundary length $2p$ in the non-generic critical regime, showing $V/\overline{V}(p) \to W_\infty$ for $n\in(0,2)$ and $(\ln p)^{-1} V/\overline{V}(p) \to D_\infty$ for $n=2$, with $W_\infty$ described by the Chen–Curien–Maillard multiplicative cascade and $D_\infty$ by its derivative martingale; in the dilute case $W_\infty$ matches the area of a unit-boundary $\gamma$-quantum disc, linking discrete maps to Liouville quantum gravity. The proof combines the gasket decomposition, BDG/Janson–Stefánsson encodings, a spine Markov chain, and a robust good/bad region classification, together with discrete-to-continuum martingale convergence and barrier arguments to handle the boundary case. This work confirms key physics conjectures about the continuum limits of loop models on random planar maps and provides a rigorous bridge between combinatorial map ensembles and continuum quantum gravity objects such as the $\gamma$-quantum disc and related CLE constructions. The results thus offer a rigorous probabilistic foundation for the scaling limits of loop-$O(n)$ maps in two-dimensional quantum gravity and illuminate the nature of critical geometries in these models.

Abstract

We study the volume of rigid loop-$O(n)$ quadrangulations with a boundary of length $2p$ in the non-generic critical regime. We prove that, as the half-perimeter $p$ goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen, Curien and Maillard arXiv:1702.06916, or alternatively (in the dilute case) as the law of the area of a unit-boundary $γ$-quantum disc, as determined by Ang and Gwynne arXiv:1903.09120, for suitable $γ$. Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade. We stress that our techniques enable us to include the boundary case $n=2$, that we define rigorously, and where the nested cascade structure is that of a critical branching random walk. In that case the scaling limit is given by the limit of the derivative martingale and is inverse-exponentially distributed, which answers a conjecture of arXiv:2005.06372v2.

Figures (10)

• Figure 1: The two types of faces of a loop-decorated quadrangulation $(\mathfrak q,\bf \ell)$, with corresponding weights $g$ and $h$. Each loop receives additional (global) weight $n$.
• Figure 2: A loop-decorated quadrangulation $(\mathfrak q,\bf \ell)$. The boundary of the map is shown in bold, with a distinguished oriented root edge. The outermost loops are shown in blue, one interior loop is shown in red.
• Figure 3: Phase diagram of the $O(n)$ model on quadrangulations borot2011recursivebudd2018peeling for fixed $n\in(0,2)$ (the diagram looks roughly the same for all $n$). The critical line separates the non-admissible region (where the partition function \ref{['eq: partition function O(n)']} blows up) from the subcritical regime, where the maps are believed to yield trees in the limit. On the critical line, interesting behaviours are expected, with convergence either to the Brownian disk in the generic critical regime (red), or other objects in the non-generic critical regime (blue). The latter can be further split into dense (blue line) or dilute (blue point) phases, where $\alpha := \frac{3}{2} - \frac{1}{\pi} \arccos(n/2)$ or $\alpha := \frac{3}{2} + \frac{1}{\pi} \arccos(n/2)$in \ref{['eq: relation alpha and n']} respectively.
• Figure 4: The gasket decomposition of the loop-decorated quadrangulation $(\mathfrak q,\bf\ell)$ in \ref{['fig: loop-decorated quadrangulation']}. On the left is the gasket $\mathfrak g$, whereas on the right are the other connected components. All of these come with a deterministic choice of root edge, that we did not represent. To be precise, we stress that the gluing operation also requires rooted holes (this is discussed thoroughly in borot2011recursive).
• Figure 5: The Bouttier--Di Francesco--Guitter bijection. Left: the planar map $\mathfrak m$ is represented in blue, with its marked (square) vertex $\rho$. We draw additional (dashed) edges between a vertex and a face according to Step 2. Right: The forest obtained by disconnecting the external vertex $v_{\textnormal{ext}}$ -- the number of trees corresponds to the half-perimeter of $\mathfrak m$.

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1
• Proposition 2.1
• Remark 2.2
• Theorem 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• proof
• Proposition 2.7