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

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• Polymers and percolation in two dimensions and twisted N=2 supersymmetry
• Conformal Field Theories of Stochastic Loewner Evolutions
• Conformally Invariant Fractals and Potential Theory
• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• Path Crossing Exponents and the External Perimeter in 2D Percolation
• Multicritical Phases of the O(n) Model on a Random Lattice
• Euler integrals for commuting SLEs
• Conformal Field Theory Techniques in Random Matrix models
• SLE martingales and the Virasoro algebra
• Excursion decompositions for $\SLE$ and Watts' crossing formula
• Exact Multifractal Exponents for Two-Dimensional Percolation
• Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model
• Boundary Liouville Theory and 2D Quantum Gravity
• Dipolar SLEs
• On Conformal Field Theory and Stochastic Loewner Evolution
• Calogero-Sutherland model and bulk-boundary correlations in conformal field theory
• Boundary Correlators in 2D Quantum Gravity: Liouville versus Discrete Approach
• Loop Gas Model for Open Strings
• Conformal Fractal Geometry and Boundary Quantum Gravity
• Two-Dimensional Copolymers and Exact Conformal Multifractality
• Meanders: Exact Asymptotics
• Conformal Invariance and Percolation
• The Scaling Limit Geometry of Near-Critical 2D Percolation
• The Full Scaling Limit of Two-Dimensional Critical Percolation
• Fractal geometry of critical Potts clusters
• Harmonic Measure and Winding of Conformally Invariant Curves
• Boundary Ground Ring in 2D String Theory
• CFTs of SLEs: the radial case
• Conformational Entropy of Compact Polymers
• On harmonic measure of critical curves
• On conformal field theory of SLE(kappa; rho)
• Continuous melting of compact polymers
• Stochastic Loewner Evolution and Dyson's Circular Ensembles
• Integrable O(n) model on the honeycomb lattice via reflection matrices : Surface critical behaviour
• Area distribution of the planar random loop boundary
• SLE(kappa,rho) and Conformal Field Theory
• Scaling limit of loop erased random walk - a naive approach
• On Connections of Conformal Field Theory and Stochastic Lœwner Evolution
• Exact characterization of O(n) tricriticality in two dimensions
• SLE, CFT and zig-zag probabilities
• Exact results for the universal area distribution of clusters in percolation, Ising and Potts models
• Higher Conformal Multifractality
• Classification of Conformal Field Theories Based on Coulomb gases. Application to Loop Models
• Transition from the compact to the dense phase of two-dimensional polymers
• Fractal dimensions and corrections to scaling for critical Potts clusters

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• The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders
• Boundary Partitions in Trees and Dimers
• Invitation to Random Tensors
• The Double Scaling Limit in Arbitrary Dimensions: A Toy Model
• Many-Body-Localization Transition : strong multifractality spectrum for matrix elements of local operators
• Geometric Exponents, SLE and Logarithmic Minimal Models
• Numerical study on Schramm-Loewner Evolution in nonminimal conformal field theories
• Renormalizable Models in Rank $d\ge 2$ Tensorial Group Field Theory
• Two-point connectivity of two-dimensional critical $Q-$ Potts random clusters on the torus
• Harmonic measure and winding of random conformal paths: A Coulomb gas perspective
• Liouville Quantum Gravity and KPZ
• Directed polymer in a random medium of dimension 1+3 : multifractal properties at the localization/delocalization transition
• Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus
• On the multifractal statistics of the local order parameter at random critical points : application to wetting transitions with disorder
• Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties
• Pure and Random Quantum Ising Chain : Shannon and Renyi entropies of the ground state via real space renormalization
• Random cascade models of multifractality : real-space renormalization and travelling-waves
• The Coefficient Problem and Multifractality of Whole-Plane SLE and LLE
• Generalized disconnection exponents
• Level Set Percolation in Two-Dimensional Gaussian Free Field
• Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points
• Connecting SLE and minisuperspace Liouville gravity