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

### Linxiao Chen

ETH Zürich, Switzerland### Nicolas Curien

Université Paris-Sud, Université Paris-Saclay, Orsay, France### Pascal Maillard

Université de Toulouse III – Paul Sabatier, Toulouse, France

## 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 $(x_{i})_{i≥1}$ is related to the jumps of a spectrally positive $α$-stable Lévy process with $α=23 ±π1 arccos(n/2)$ 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.

## Cite this article

Linxiao Chen, Nicolas Curien, Pascal Maillard, The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 7 (2020), no. 4, pp. 535–584

DOI 10.4171/AIHPD/94