The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Annales de l’Institut Henri Poincaré D

Full-Text PDF (679 KB) | Metadata | Table of Contents | AIHPD summary
Volume 7, Issue 4, 2020, pp. 535–584
DOI: 10.4171/AIHPD/94

Published online: 2020-11-30

The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model

Linxiao Chen[1], Nicolas Curien[2] and Pascal Maillard[3]

(1) ETH Zürich, Switzerland
(2) Université Paris-Sud, Université Paris-Saclay, Orsay, France
(3) Université de Toulouse III – Paul Sabatier, Toulouse, France

We study the branching tree of the perimeters of the nested loops in the non-generic critical $O(n)$ model on random quadrangulations. We prove that after renormalization it converges towards an explicit continuous multiplicative cascade whose offspring distribution $(x_i)_{i \geq 1}$ is related to the jumps of a spectrally positive $\alpha$-stable Lévy process with $\alpha= \frac{3}{2} \pm \frac{1}{\pi} \mathrm {arccos}(n/2)$ and for which we have the surprisingly simple and explicit transform $$ \mathbb{E}\Big[ \sum_{i \geq 1}(x_i)^\theta \Big] = \frac{\sin(\pi (2-\alpha))}{\sin (\pi (\theta - \alpha))}, \quad \mbox{for }\theta \in (\alpha, \alpha+1).$$ An important ingredient in the proof is a new formula of independent interest on first moments of additive functionals of the jumps of a left-continuous random walk stopped at a hitting time. We also identify the scaling limit of the volume of the critical $O(n)$-decorated quadrangulation using the Malthusian martingale associated to the continuous multiplicative cascade.

Keywords: $O(n)$ loop model, random planar map, multiplicative cascade, invariance principle, stable Lévy process

Chen Linxiao, Curien Nicolas, Maillard Pascal: The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 7 (2020), 535-584. doi: 10.4171/AIHPD/94