Conformal Random Geometry

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents associated with interacting random paths, by exploiting their underlying quantum gravity structure. The latter relates exponents in the plane to those on a random lattice, i.e., in a fluctuating metric, using the so-called Knizhnik, Polyakov and Zamolodchikov (KPZ) map. This is accomplished within the framework of random matrix theory and conformal field theory, with applications to geometrical critical models, like Brownian paths, self-avoiding walks, percolation, and more generally, the O(N) or Q-state Potts models and, last but not least, Schramm's Stochastic Loewner Evolution (SLE_kappa). These Notes can be considered as complementary to those by Wendelin Werner (2006 Fields Medalist!), ``Some Recent Aspects of Random Conformally Invariant Systems,'' arXiv:math.PR/0511268.