Conformal Fractal Geometry and Boundary Quantum Gravity

Abstract

This article gives a comprehensive description of the fractal geometry of conformally-invariant (CI) scaling curves, in the plane or half-plane. It focuses on deriving critical exponents associated with interacting random paths, by exploiting an underlying quantum gravity (QG) structure, which uses KPZ maps relating exponents in the plane to those on a random lattice, i.e., in a fluctuating metric. This is applied to critical models, like O(N) and Potts models, and to the Stochastic Löwner Evolution (SLE). The multifractal (MF) function f(alpha, c) of the harmonic measure near any CI fractal boundary, is given as a function of the central charge c of the associated CFT. The Hausdorff dimensions D_{H} of a non-simple scaling curve or cluster hull, and D_{EP} of its external perimeter or frontier, are shown to obey the duality equation (D_{H}-1)(D_{EP}-1)=1/4, valid for any c. The universal mixed MF spectrum f(alpha,lambda;c) describing the local spiralling rate lambda and singularity exponent alpha of the potential near any CI scaling curve is given. The duality between simple and non-simple random paths is established via a symmetry of the KPZ quantum gravity map. An extended dual KPZ relation is introduced for the SLE_{kappa}, which commutes with the kappa to kappa'=16/kappa duality. This gives the SLE exponents from simple QG rules, established from the general structure of correlation functions of arbitrary interacting random sets on a random lattice.

118 pages, 36 figures, research review article. The first part (Sections 2-7) is an expanded version of J. Stat. Phys. 110, 691 (2003), cond-mat/0207743. The second part (sections 8-12 and appendices) focuses on SLE and quantum gravity, and on their various dualities, mirrored in that relating simple and non-simple paths. Finally, it offers detailed arguments leading to the quantum gravity approach to interacting random paths

References in corpus (7)

• Conformal Field Theories of Stochastic Loewner Evolutions
• SLE martingales and the Virasoro algebra
• Boundary Correlators in 2D Quantum Gravity: Liouville versus Discrete Approach
• Harmonic Measure and Winding of Conformally Invariant Curves
• Stochastic Loewner Evolution and Dyson's Circular Ensembles
• Higher Conformal Multifractality
• Fractal dimensions and corrections to scaling for critical Potts clusters

Cited by in corpus (27)

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• Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
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• The Loewner equation: maps and shapes
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• Conformal Random Geometry
• Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model
• Boundary Liouville Theory and 2D Quantum Gravity
• Brownian Motion, Chern-Simons Theory, and 2d Yang-Mills
• Random walks on combs
• Fractal Structure of Spin Clusters and Domain Walls in 2D Ising Model
• Topics in coarsening phenomena
• CFTs of SLEs: the radial case
• Gaussian multiplicative chaos and applications: a review
• Anchored Critical Percolation Clusters and 2-D Electrostatics
• The multifractal spectrum of Brownian intersection local times
• Local logarithmic correlators as limits of Coulomb gas integrals
• Almost sure multifractal spectrum of SLE
• Harmonic measure and winding of random conformal paths: A Coulomb gas perspective
• Liouville Quantum Gravity and KPZ
• Statistics of harmonic measure and winding of critical curves from conformal field theory
• Nesting statistics in the O(n) loop model on random planar maps
• The density of critical percolation clusters touching the boundaries of strips and squares
• Multiple-SLE connectivity weights for rectangles, hexagons, and octagons
• Alternating Arm Exponents for the Critical Planar Ising Model
• Classical and Quantum Integrability in Laplacian Growth