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 $(x_i{)}_{i\ge 1}$ is related to the jumps of a spectrally positive $\alpha$-stable Lévy process with $\alpha =\frac{3}{2}\pm \frac{1}{\pi }\arccos (n/2)$ and for which we have the surprisingly simple and explicit transform

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.