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# Annales de l’Institut Henri Poincaré D

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**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