Random planar curves and Schramm-Loewner evolutions

Abstract

We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.

(Final?) version of the Notes of the lectures delivered at the Saint-Flour summer school (July 2002). 100 pages, 16 figures. To appear in Springer L.N

References in corpus (12)

• The dimension of the SLE curves
• Local statistics of lattice dimers
• Conformal Field Theories of Stochastic Loewner Evolutions
• SLE_k growth processes and conformal field theories
• SLE martingales and the Virasoro algebra
• The degree distribution in bipartite planar maps: applications to the Ising model
• Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk
• Conformal Fractal Geometry and Boundary Quantum Gravity
• Brownian beads
• Scaling limit of loop erased random walk - a naive approach
• The lowest crossing in 2D critical percolation
• The Brownian loop soup

Cited by in corpus (58)

• 2D growth processes: SLE and Loewner chains
• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• SLE coordinate changes
• Euler integrals for commuting SLEs
• The Loewner equation: maps and shapes
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• Calogero-Sutherland model and bulk-boundary correlations in conformal field theory
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Fusion algebra of critical percolation
• On Multiple Schramm-Loewner Evolutions
• Convergence of the two-dimensional random walk loop soup clusters to CLE
• Stochastic Loewner evolution for conformal field theories with Lie-group symmetries
• Multi-point Green's functions for SLE and an estimate of Beffara
• LERW as an example of off-critical SLEs
• Twenty-five years of two-dimensional rational conformal field theory
• The expected area of the filled planar Brownian loop is Pi/5
• Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
• Corner percolation on ${\mathbb{Z}}^2$ and the square root of 17
• The Loewner energy of loops and regularity of driving functions
• On conformal field theory of SLE(kappa; rho)
• Scaling of Percolation on Infinite Planar Maps, I
• Loop-erased random walk and Poisson kernel on planar graphs
• Critical percolation in annuli and $SL{E}_6$
• Ising (Conformal) Fields and Cluster Area Measures
• Quantum Hall transitions: An exact theory based on conformal restriction
• Geometric Exponents, SLE and Logarithmic Minimal Models
• SLE local martingales in logarithmic representations
• Proposal for a CFT interpretation of Watts' differential equation for percolation
• The scaling limit of two cluster boundaries in critical lattice models
• The Hausdorff dimension of the CLE gasket
• Almost sure multifractal spectrum of SLE
• Area distribution of the planar random loop boundary
• Universality for SLE(4)
• SLE(kappa,rho) and Conformal Field Theory
• SLE(6) and CLE(6) from Critical Percolation
• Using the Schramm-Loewner evolution to explain certain non-local observables in the 2d critical Ising model
• Conformal welding problem, flow line problem, and multiple Schramm--Loewner evolution
• SLE local martingales, reversibility and duality
• Large deviations of multichordal SLE${}_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians
• Random loops and conformal field theory
• Monte Carlo comparisons of the self-avoiding walk and SLE as parameterized curves
• Reflected Brownian motion in generic triangles and wedges
• The trunks of CLE(4) explorations
• The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics
• The scaling limit of Fomin's identity for two paths in the plane
• Local martingales associated with Schramm-Loewner evolutions with internal symmetry
• Stochastic Komatu-Loewner evolutions and BMD domain constant
• Hypotrochoids in conformal restriction systems and Virasoro descendants
• SLE and alpha-SLE driven by Levy processes
• Loewner chains on the universal covering space of a Riemann surface
• Numerical studies of planar closed random walks
• On Malliavin measures, SLE and CFT
• Towards Conformal Invariance and a Geometric Representation of the 2D Ising Magnetization Field
• Brownian motion correlation in the peanosphere for $\kappa >8$
• Connection probabilities in the double-dimer model -- the case of two connectivity patterns
• The expectation value of the number of loops and the left-passage probability in the double-dimer model
• Stochastic Loewner Evolution
• Numerical evaluation of the Gauss hypergeometric function: Implementation and application to Schramm-Loewner evolution