Conformal Random Geometry

Abstract

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.

116 pages, 45 figures. Les Houches 2005 Lecture Notes. Based on the previous research survey article ``Conformal Fractal Geometry and Boundary Quantum Gravity,'' arXiv:math-ph/0303034 [in: ``Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot'' (M. L. Lapidus and M.van Frankenhuysen, eds.), Proc.Symposia Pure Math. vol. 72, Part 2, 365-482 (AMS, Providence, R.I., 2004], here augmented by introductory sections, explanatory figures and some new material. Supplementary technical appendices can be found in the foregoing reference

References in corpus (7)

• Conformal Fractal Geometry and Boundary Quantum Gravity
• The Full Scaling Limit of Two-Dimensional Critical Percolation
• Continuous melting of compact polymers
• SLE(kappa,rho) and Conformal Field Theory
• Scaling limit of loop erased random walk - a naive approach
• SLE, CFT and zig-zag probabilities
• Exact results for the universal area distribution of clusters in percolation, Ising and Potts models

Cited by in corpus (5)

• Numerical study on Schramm-Loewner Evolution in nonminimal conformal field theories
• Harmonic measure and winding of random conformal paths: A Coulomb gas perspective
• Directed polymer in a random medium of dimension 1+3 : multifractal properties at the localization/delocalization transition
• On the multifractal statistics of the local order parameter at random critical points : application to wetting transitions with disorder
• Connecting SLE and minisuperspace Liouville gravity