A Glimpse of the Conformal Structure of Random Planar Maps

Nicolas Curien (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm–Loewner evolution) process of parameter κ=6 and to combine the locality property of the SLE6 together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary. Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension 13 almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF).

Domaines: Physique [physics]

Nicolas Curien. A Glimpse of the Conformal Structure of Random Planar Maps. Communications in Mathematical Physics, 2014, 333 (3), pp.1417-1463. ⟨10.1007/s00220-014-2196-5⟩. ⟨hal-01122764⟩