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

Abstract

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 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.