Percolation on random triangulations and stable looptrees

Abstract

We study site percolation on Angel & Schramm's uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3/2, which was introduced in a companion paper. We also give a conjecture linking looptrees of any index in (1,2) with scaling limits of cluster boundaries in random triangulations decorated with O(N) models.

30 pages, 12 figures. Published version (Probab. Theory Related Fields)

References in corpus (6)

• Uniqueness and universality of the Brownian map
• Local limit of labeled trees and expected volume growth in a random quadrangulation
• Random stable looptrees
• Scaling limits of random planar maps with a unique large face
• Condensation in nongeneric trees
• More on the O(n) model on random maps via nested loops: loops with bending energy

Cited by in corpus (4)

• Random stable looptrees
• The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model
• Geometry and percolation on half planar triangulations
• Sub-exponential tail bounds for conditioned stable Bienaymé-Galton-Watson trees