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

Chen, Linxiao; Curien, Nicolas; Maillard, Pascal

doi:10.4171/aihpd/94

The perimeter cascade in critical Boltzmann quadrangulations decorated by an

O (n)

loop model

Chen, Linxiao ; Curien, Nicolas ; Maillard, Pascal

Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 4, pp. 535-584.

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Résumé

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 $α$ -stable Lévy process with $α = \frac{3}{2} \pm \frac{1}{π} \arccos (n / 2)$ and for which we have the surprisingly simple and explicit transform

𝔼 [\sum_{i \geq 1} {(x_{i})}^{θ}] = \frac{sin (π (2 - α))}{sin (π (θ - α))}, f o r θ \in (α, α + 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.

Accepté le : 2019-06-28
Publié le : 2020-11-30

MR Zbl

DOI : 10.4171/aihpd/94

Classification : 60-XX, 05-XX
Mots-clés : $O(n)$ loop model, random planar map, multiplicative cascade, invariance principle, stable Lévy process

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     title = {The perimeter cascade in critical {Boltzmann} quadrangulations decorated by an $O(n)$ loop model},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {535--584},
     volume = {7},
     number = {4},
     year = {2020},
     doi = {10.4171/aihpd/94},
     mrnumber = {4182775},
     zbl = {1454.05106},
     language = {en},
     url = {http://archive.numdam.org/articles/10.4171/aihpd/94/}
}

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Chen, Linxiao; Curien, Nicolas; Maillard, Pascal. The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model. Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 4, pp. 535-584. doi : 10.4171/aihpd/94. http://archive.numdam.org/articles/10.4171/aihpd/94/

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