2D growth processes: SLE and Loewner chains

Abstract

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE. This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.

A review on Stochastic Loewner evolutions for Physics Reports, 172 pages, low quality figures, better quality figures upon request to the authors, comments welcome

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• A Guide to Stochastic Loewner Evolution and its Applications
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• The Loewner equation: maps and shapes
• Conformal Fractal Geometry and Boundary Quantum Gravity
• SLE(κ,ρ) martingales and duality
• Stochastic Loewner evolution for conformal field theories with Lie-group symmetries
• Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy
• Critical percolation in annuli and $SL{E}_6$
• Diffusion-limited aggregation in channel geometry
• SLE(kappa,rho) and Conformal Field Theory
• On radial stochastic Loewner evolution in multiply connected domains
• Effects of stochastic nucleation in the first order phase transition

Cited by in corpus (41)

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• Conformal Curves on $W{O}_3$ Surface
• Critical curves in conformally invariant statistical systems
• SLE in the three-state Potts model - a numerical study
• Conformal Invariance of Iso-height Lines in two-dimensional KPZ Surface
• Fingered growth in channel geometry: A Loewner equation approach
• Thermal Behavior of Spin Clusters and Interfaces in two-dimensional Ising Model on Square Lattice
• First passage time processes and subordinated SLE
• Virasoro Module Structure of Local Martingales of SLE Variants
• Scaling of Clusters and Winding Angle Statistics of Iso-height Lines in two-dimensional KPZ Surface
• Domain walls and chaos in the disordered SOS model
• SLE in self-dual critical Z(N) spin systems: CFT predictions
• Geometric Exponents, SLE and Logarithmic Minimal Models
• Exact solution of the anisotropic special transition in the O(n) model in 2D
• SLE local martingales in logarithmic representations
• Global properties of Stochastic Loewner evolution driven by Levy processes
• Numerical study on Schramm-Loewner Evolution in nonminimal conformal field theories
• Critical behavior of interfaces in disordered Potts ferromagnets : statistics of free-energy, energy and interfacial adsorption
• Towards conformal invariance of 2D lattice models
• Harmonic measure and winding of random conformal paths: A Coulomb gas perspective
• Percolation and Schramm-Loewner evolution in the 2D random-field Ising model
• Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model
• Statistics of harmonic measure and winding of critical curves from conformal field theory
• Disordered O(n) Loop Model and Coupled Conformal Field Theories
• SLE local martingales, reversibility and duality
• Conformal loop ensembles and the stress-energy tensor. II. Construction of the stress-energy tensor
• Geometric and Stochastic Clusters of Gravitating Potts Models
• Conformal loop ensembles and the stress-energy tensor. I. Fundamental notions of CLE
• SLE on doubly-connected domains and the winding of loop-erased random walks
• A generalisation of Schramm's formula for SLE(2)
• Stochastic Process Associated with Traveling Wave Solutions of the Sine-Gordon Equation
• Loop models for CFTs
• Loewner driving functions for off-critical percolation clusters
• A Renormalisation Group approach to Stochastic Lœwner Evolutions and the Doob h-transform
• Intersection probabilities for a chordal SLE path and a semicircle
• Stochastic Loewner Evolution
• $SLE(\kappa ,\rho )$ processes, hiding exponents and self-avoiding walks in a wedge
• Connecting SLE and minisuperspace Liouville gravity
• CFT Interpretation of Merging Multiple SLE Traces