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

## TL;DR

This paper analyzes the nested loop perimeters in critical $O(n)$ models on random quadrangulations, showing convergence to a continuous multiplicative cascade related to a spectrally positive $rac{3}{2} ext{-}rac{1}{ ext{pi}} ext{arccos}(n/2)$-stable Lévy process, and identifies the volume scaling limit.

## Contribution

It introduces an explicit continuous multiplicative cascade model for the perimeter structure and derives a new formula for moments of additive functionals of jump processes.

## Key findings

- Convergence of perimeter trees to a multiplicative cascade.
- Explicit formula for offspring distribution related to a stable Lévy process.
- Identification of the volume scaling limit using a Malthusian martingale.

## Abstract

We study the branching tree of the perimeters of the nested loops in critical $O(n)$ model for $n \in (0,2)$ 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 L\'evy process with $\alpha= \frac{3}{2} \pm \frac{1}{\pi} \arccos(n/2)$ and for which we have the surprisingly simple and explicit transform $$ \mathbb E\left[\sum_{i \ge 1} (x_i)^\theta \right] = \frac{\sin(\pi (2-\alpha))}{\sin (\pi (\theta - \alpha))} \quad \mbox{for }\theta \in (\alpha, \alpha+1).$$ 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.