Résumé |
We study the volume of rigid loop-O(n) quadrangulations with a boundary of length
2p in the critical non-generic regime. We prove that, as the half-perimeter p goes
to infinity, the volume scales in distribution to an explicit random variable.
This limiting random variable is described in terms of the multiplicative cascades
of Chen, Curien and Maillard, or alternatively (in the dilute case) as the law of
the area of a suitable unit-boundary quantum disc, as determined by Ang and
Gwynne. Our arguments go through a classification of the map into several regions,
where we rule out the contribution of bad regions to be left with a tractable
portion of the map. One key observable for this classification is a Markov chain
which explores the nested loops around a size-biased vertex pick in the map,
making explicit the spinal structure of the discrete multiplicative cascade. This
talk is based on joint work with Élie Aïdékon and XingJian Hu (Fudan University). |