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The perimeter cascade in critical Boltzmann quadrangulations decorated by an \(O(n)\) loop model. (English) Zbl 1454.05106

Summary: 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 \(\alpha= \frac{3}{2} \pm \frac{1}{\pi} \text{arccos}(n/2)\) and for which we have the surprisingly simple and explicit transform \[ \mathbb{E} \left[ \sum\limits_{i \geq 1}(x_i)^\theta \right] = \frac{\sin(\pi (2 - \alpha))}{\sin (\pi (\theta - \alpha))}, \quad \text{for } \theta \in (\alpha, \alpha + 1). \]
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.

MSC:

05C80 Random graphs (graph-theoretic aspects)
05C81 Random walks on graphs
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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