A Glimpse of the Conformal Structure of Random Planar Maps

作者：Curien Nicolas^*

来源：Communications in Mathematical Physics, 2015, 333(3): 1417-1463.

DOI：10.1007/s00220-014-2196-5

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 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 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).

• 出版日期2015-2