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 $(x_i{)}_{i\ge 1}$ is related to the jumps of a spectrally positive $\alpha$-stable Lévy process with $\alpha =\frac{3}{2}\pm \frac{1}{\pi }\arccos (n/2)$ and for which we have the surprisingly simple and explicit transform $$ \mathbb E\left[\sum_{i \ge 1} (x_i)^θ\right] = \frac{\sin(π(2-α))}{\sin (π(θ- α))} \quad \mbox{for }θ\in (α, α+1).$$ 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.

35 pages, 7 figures, minor adjustments

References in corpus (3)

• Conformal radii for conformal loop ensembles
• Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $\kappa >8$
• Nesting statistics in the O(n) loop model on random planar maps

Cited by in corpus (8)

• Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond
• The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen)
• Nesting statistics in the O(n) loop model on random planar maps
• Local convergence of large random triangulations coupled with an Ising model
• Planar maps and random partitions
• The exploration process of critical Boltzmann planar maps decorated by a triangular $O(n)$ loop model
• Limits of the boundary of random planar maps
• Liouville quantum gravity weighted by conformal loop ensemble nesting statistics