Orateur
Pascal Maillard
(Université Paris-Sud Orsay)
Description
One approach to understanding Liouville Quantum Gravity is to study random planar maps endowed with a statistical mechanics model. In this talk, I will present a work in progress with Nicolas Curien and Linxiao Chen on a certain model of random loops on random quadrangulations: the critical $O(n)$ model for $n \in (0,2)$. We prove that after renormalization the branching tree of the perimeters of the nested loops 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 $\alpha= \frac{3}{2} \pm \frac{1}{\pi} \arccos(n/2)$ and for which we can compute explicitly the Mellin transform (a.k.a. the Laplace transform in the context of branching random walks).
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, which might be of independent interest.
Auteur principal
Pascal Maillard
(Université Paris-Sud Orsay)
