Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory

Abstract

Conformally-invariant curves that appear at critical points in two-dimensional statistical mechanics systems, and their fractal geometry have received a lot of attention in recent years. On the one hand, Schramm has invented a new rigorous as well as practical calculational approach to critical curves, based on a beautiful unification of conformal maps and stochastic processes, and by now known as Schramm-Loewner evolution (SLE). On the other hand, Duplantier has applied boundary quantum gravity methods to calculate exact multifractal exponents associated with critical curves. In the first part of this paper I provide a pedagogical introduction to SLE. I present mathematical facts from the theory of conformal maps and stochastic processes related to SLE. Then I review basic properties of SLE and provide practical derivation of various interesting quantities related to critical curves, including fractal dimensions and crossing probabilities. The second part of the paper is devoted to a way of describing critical curves using boundary conformal field theory (CFT) in the so-called Coulomb gas formalism. This description provides an alternative (to quantum gravity) way of obtaining the multifractal spectrum of critical curves using only traditional methods of CFT based on free bosonic fields.

arXiv commentPublished version, references added and rearranged, typos corrected

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• A solution space for a system of null-state partial differential equations 3
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• Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
• Virasoro Module Structure of Local Martingales of SLE Variants
• A solution space for a system of null-state partial differential equations 1
• Scaling of Clusters and Winding Angle Statistics of Iso-height Lines in two-dimensional KPZ Surface
• A solution space for a system of null-state partial differential equations 4
• Quantum Hall transitions: An exact theory based on conformal restriction
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• The Harmonic Measure of Diffusion-Limited Aggregates including Rare Events
• Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background
• The Generic Critical Behaviour for 2D Polymer Collapse
• Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model
• Critical interfaces of the Ashkin-Teller model at the parafermionic point
• Statistics of harmonic measure and winding of critical curves from conformal field theory
• On SLE martingales in boundary WZW models
• First passage times and distances along critical curves
• Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions
• Stochastic geometry and topology of non-Gaussian fields
• On Harmonic Measure of the Whole Plane Levy-Loewner Evolution
• Random loops and conformal field theory
• On Exact Multi-fractal Spectrum of the Whole-Plane SLE
• SLE_k: correlation functions in the coefficient problem
• Cluster pinch-point densities in polygons
• The scaling limit of Fomin's identity for two paths in the plane
• Whole-plane self-avoiding walks and radial Schramm-Loewner evolution: a numerical study
• Conformal Invariance in (2+1)-Dimensional Stochastic Systems
• Hypotrochoids in conformal restriction systems and Virasoro descendants
• q-deformed Loewner evolution
• On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth
• Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling
• The Harmonic Measure for critical Potts clusters
• Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model
• Loewner driving functions for off-critical percolation clusters
• Stochastic Loewner Evolutions, Fuchsian Systems and Orthogonal Polynomials
• Geometrically Regulating Evolutionary Dynamics in Biofilms
• Connecting SLE and minisuperspace Liouville gravity
• Martingales of stochastic Laplacian growth