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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 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 $(x_i)_{i \geq 1}$ 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 can compute explicitly the transform $$ \mathbb{E}\left[ \sum_{i \geq 1}(x_i)^\theta \right] = \frac{\sin(\pi (2-\alpha))}{\sin (\pi (\theta - \alpha))} \quad \mbox{for }\theta \in (\alpha, \alpha+1).$$ An important ingredient in the proof is a new formula on first moments of additive functionals of the jumps of a left-continuous random walk stopped at a hitting time.
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Contributor : Nicolas Curien <>
Submitted on : Thursday, July 15, 2021 - 3:46:11 PM
Last modification on : Monday, July 26, 2021 - 4:28:04 PM

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Linxiao Chen, Nicolas Curien, Pascal Maillard. The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model. Annales de l’Institut Henri Poincaré (D) Combinatorics, Physics and their Interactions, European Mathematical Society, 2020, 7 (4), pp.535-584. ⟨10.4171/aihpd/94⟩. ⟨hal-03287391⟩

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