A Guide to Stochastic Loewner Evolution and its Applications

Abstract

This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Loewner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Loewner's method, explaining how this method can be used to describe families of random curves. Then we define SLE and discuss some of its properties. We also explain how the connection can be made between SLE and the discrete models whose scaling limits it describes, or is believed to describe. Finally, we have included a discussion of results that were obtained from SLE computations. Some explicit proofs are presented as typical examples of such computations. To understand SLE sufficient knowledge of conformal mapping theory and stochastic calculus is required. This material is covered in the appendices.

80 pages, 22 figures, LaTeX; this version has 5 minor corrections to the text and improved hyperref support

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• Conformally Invariant Fractals and Potential Theory
• Values of Brownian intersection exponents III: Two-sided exponents
• SLE_k growth processes and conformal field theories
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Cited by in corpus (96)

• Recent advances in percolation theory and its applications
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• Out-of-equilibrium dynamics of quantum many-body systems with long-range interactions
• Time evolution of 1D gapless models from a domain-wall initial state: SLE continued?
• The two-boundary Temperley-Lieb algebra
• Nodal portraits of quantum billiards: Domains, lines, and statistics
• Fusion algebra of critical percolation
• Metal-Insulator Transition in Graphene on Boron Nitride
• SLE description of the nodal lines of random wave functions
• Numerical computations for the Schramm-Loewner Evolution
• Critical curves in conformally invariant statistical systems
• Cluster simulations of loop models on two-dimensional lattices
• Towards physical principles of biological evolution
• Some geometric critical exponents for percolation and the random-cluster model
• Fingered growth in channel geometry: A Loewner equation approach
• LERW as an example of off-critical SLEs
• Worm Monte Carlo study of the honeycomb-lattice loop model
• Thermal Behavior of Spin Clusters and Interfaces in two-dimensional Ising Model on Square Lattice
• Geometric structure of percolation clusters
• First passage time processes and subordinated SLE
• Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
• The O(n) loop model on a three-dimensional lattice
• Conformal mapping methods for interfacial dynamics
• Virasoro Module Structure of Local Martingales of SLE Variants
• Critical speeding-up in a local dynamics for the random-cluster model
• Direct Evidence for Conformal Invariance of Avalanche Frontier in Sandpile Models
• A solution space for a system of null-state partial differential equations 1
• Logarithmic conformal invariance in the Abelian sandpile model
• A formula for crossing probabilities of critical systems inside polygons
• SLE in self-dual critical Z(N) spin systems: CFT predictions
• Laplacian networks: growth, local symmetry and shape optimization
• Percolation in the canonical ensemble
• Quantum Hall transitions: An exact theory based on conformal restriction
• Geometric Exponents, SLE and Logarithmic Minimal Models
• Stochastic Loewner evolution driven by Levy processes
• Fractal nature of high-order time crystal phases
• SLE local martingales in logarithmic representations
• Random walk loop soups and conformal loop ensembles
• Global properties of Stochastic Loewner evolution driven by Levy processes
• Connection probabilities for conformal loop ensembles
• Numerical study on Schramm-Loewner Evolution in nonminimal conformal field theories
• The Hausdorff dimension of the CLE gasket
• Loop-Cluster Coupling and Algorithm for Classical Statistical Models
• Harmonic measure and winding of random conformal paths: A Coulomb gas perspective
• Note on SLE and logarithmic CFT
• Boundary correlations in planar LERW and UST
• Extreme nesting in the conformal loop ensemble
• Entanglement renormalization and boundary critical phenomena
• Recursive Percolation
• Geometrical properties of parafermionic spin models
• The conformal loop ensemble nesting field
• Discrete holomorphic parafermions in the Ashkin-Teller model and SLE
• Fractal Structure of Isothermal Lines and Loops on the Cosmic Microwave Background
• Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model
• Nesting statistics in the O(n) loop model on random planar maps
• Statistics of harmonic measure and winding of critical curves from conformal field theory
• Stochastic Loewner Evolution Relates Anomalous Diffusion and Anisotropic Percolation
• SLE local martingales, reversibility and duality
• Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution
• Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model
• Is negative-weight percolation compatible with SLE?
• Random loops and conformal field theory
• Interface growth in the channel geometry and tripolar Loewner evolutions
• Interface growth in two dimensions: A Loewner-equation approach
• Random Fractal Ansatz for the configurations of Two-Dimensional Critical Systems
• Critical domain walls in the Ashkin-Teller model
• Active spanning trees and Schramm-Loewner evolution
• Noncommutative Brownian motion
• Geometric properties of the complete-graph Ising model in the loop representation
• Extremal distance and conformal radius of a CLE_4 loop
• Divergence of the correlation length for critical planar FK percolation with $1\le q\le 4$ via parafermionic observables
• Structure of Gibbs measures for planar FK-percolation and Potts models
• Exponents for Hamiltonian paths on random bicubic maps and KPZ
• Irregularity of polymer domain boundaries in two-dimensional polymer solution
• Percolation crossing probabilities in hexagons: a numerical study
• Ashkin-Teller model on the iso-radial graphs
• Whole-plane self-avoiding walks and radial Schramm-Loewner evolution: a numerical study
• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Transforming fixed-length self-avoiding walks into radial SLE_8/3
• Hypotrochoids in conformal restriction systems and Virasoro descendants
• Statistical Mechanics of Confined Polymer Networks
• q-deformed Loewner evolution
• Iterative Site Percolation on Triangular Lattice
• The fuzzy Potts model in the plane: Scaling limits and arm exponents
• Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling
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• Search for conformal invariance in compressible two-dimensional turbulence
• The expectation value of the number of loops and the left-passage probability in the double-dimer model
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• $SLE(\kappa ,\rho )$ processes, hiding exponents and self-avoiding walks in a wedge
• Hamiltonian cycles on bicolored random planar maps
• Backbone three-point correlation function in the two-dimensional Potts model