A (planar) triangulation is a graph embedded in the two-dimensional sphere such that all its faces are surrounded by three edges. Consider a random triangulation $T_n$ chosen uniformly over all triangulations of the sphere having $n$ faces. The metric structure of $T_{n}$ endowed with the graph distance has been studied in depth during recent years. In particular, Le Gall and Miermont recently proved that the metric space obtained from $T_{n}$ by re-scaling all distances by $n^{-1/4}$ converges towards a random compact metric space called “the Brownian map”. par In this talk, we will focus on another aspect of random triangulations. Indeed, $T_{n}$ can naturally be considered as a random Riemann surface and one can study its “conformal structure” which is conjectured to be strongly linked to the Gaussian free field. I will present a path to study the conformal structure of random planar maps based on their Markovian exploration by an independent SLE$_{6}$ process.

LPMA, U. Pierre et Marie Curie, Paris