We study the branching tree of the perimeters of the nested loops in the non-generic critical model on random quadrangulations. We prove that after renormalization it converges towards an explicit continuous multiplicative cascade whose offspring distribution is related to the jumps of a spectrally positive -stable Lévy process with 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 -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 loop model. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 7 (2020), no. 4, pp. 535–584DOI 10.4171/AIHPD/94