The Loewner equation: maps and shapes

Abstract

In the last few years, new insights have permitted unexpected progress in the study of fractal shapes in two dimensions. A new approach, called Schramm-Loewner evolution, or SLE, has arisen through analytic function theory and probability theory, and given a new way of calculating fractal shapes in critical phenomena, the theory of random walks, and of percolation. We present a non-technical discussion of this development aimed to attract the attention of condensed matter community to this fascinating subject.

References in corpus (25)

• The dimension of the SLE curves
• Conformal Field Theories of Stochastic Loewner Evolutions
• Conformally Invariant Fractals and Potential Theory
• Values of Brownian intersection exponents III: Two-sided exponents
• Laplacian growth as one-dimensional turbulence
• Random planar curves and Schramm-Loewner evolutions
• Path Crossing Exponents and the External Perimeter in 2D Percolation
• SLE_k growth processes and conformal field theories
• SLE martingales and the Virasoro algebra
• Exact Multifractal Exponents for Two-Dimensional Percolation
• Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo Tests
• Conformal Fractal Geometry and Boundary Quantum Gravity
• Two-Dimensional Copolymers and Exact Conformal Multifractality
• Exact Multifractal Spectra for Arbitrary Laplacian Random Walks
• Crossing Probabilities and Modular Forms
• Reflected planar Brownian motions, intertwining relations and crossing probabilities
• Harmonic Measure and Winding of Conformally Invariant Curves
• Laplacian growth with separately controlled noise and anisotropy
• Critical percolation in annuli and $SL{E}_6$
• Conformal Invariance in Percolation, Self-Avoiding Walks and Related Problems
• The Brownian loop soup
• Higher Conformal Multifractality
• Discrete Loewner evolution
• Loewner's equation in noncommutative probability
• Conformal invariance, universality, and the dimension of the Brownian frontier

Cited by in corpus (29)

• SLE for theoretical physicists
• 2D growth processes: SLE and Loewner chains
• Conformal invariance in two-dimensional turbulence
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• A Sharp Condition for the Loewner Equation to Generate Slits
• Time evolution of 1D gapless models from a domain-wall initial state: SLE continued?
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• SLE curves and natural parametrization
• Fingered growth in channel geometry: A Loewner equation approach
• Laplacian networks: growth, local symmetry and shape optimization
• Interfacial dynamics in transport-limited dissolution
• The scaling limit of two cluster boundaries in critical lattice models
• Global properties of Stochastic Loewner evolution driven by Levy processes
• Proposal for a CFT interpretation of Watts' differential equation for percolation
• Diffusion-Limited Aggregation on Curved Surfaces
• Conformal invariance in three-dimensional rotating turbulence
• First-principles analysis of nanoelectromechanical systems using Loewner equation
• Conformal invariance of weakly compressible two-dimensional turbulence
• Interface growth in the channel geometry and tripolar Loewner evolutions
• Interface growth in two dimensions: A Loewner-equation approach
• Stabilizing effect of tip splitting on the interface motion
• Trace for the Loewner Equation with Singular Forcing
• Boundary proximity of SLE
• Coastlines violate the Schramm-Loewner Evolution
• SLE($\kappa ,\rho$)and Boundary Coulomb Gas
• Loewner equation for Laplacian growth: A Schwarz-Christoffel-transformation approach
• Geodesic Loewner paths with varying boundary conditions
• Where do rivers grow? Path selection and growth in a harmonic field
• Stochastic Loewner Evolution