[Paper Review] Random planar curves and Schramm-Loewner evolutions

Wendelin Werner|arXiv (Cornell University)|Mar 27, 2003

Stochastic processes and statistical mechanicsMathematics96 references138 citations

TL;DR

This paper introduces Schramm-Loewner Evolution (SLE) as a universal scaling limit for random planar curves in two-dimensional statistical mechanics, using Loewner's differential equation driven by Brownian motion. It establishes that SLE with parameter κ=6 describes the scaling limit of critical percolation interfaces and κ=8/3 describes self-avoiding walks, providing a rigorous framework for computing critical exponents and conformal invariance in 2D systems.

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.

Motivation & Objective

• To establish Schramm-Loewner Evolution (SLE) as a universal scaling limit for random curves in two-dimensional critical statistical mechanics models.
• To connect SLE to discrete models such as loop-erased random walks and critical percolation interfaces via convergence in the scaling limit.
• To characterize SLE through key properties like locality (for SLE₆) and restriction (for SLE₈⁄₃), linking them to physical models.
• To compute critical exponents for disconnection and non-intersection probabilities using SLE, particularly showing decay like t⁻¹⁄⁸ for Brownian motion.
• To explore deeper connections between SLE, conformal field theory, and quantum gravity, especially through the KPZ relation and duality conjectures.

Proposed method

• Uses Loewner’s differential equation in the upper half-plane to define chordal SLE as a random process growing from boundary points.
• Applies stochastic calculus and Itô’s formula to analyze the evolution of conformal maps driven by a Brownian motion with parameter κ.
• Introduces radial SLE for curves growing toward an interior point, showing equivalence to chordal SLE in the κ=6 case.
• Employs the locality property of SLE₆ and restriction property of SLE₈⁄₃ to derive geometric and probabilistic results.
• Uses the link between radial SLE₆ and planar Brownian motion to compute the disconnection exponent as 1/8 via SLE techniques.
• Applies duality conjectures and ρ-parameters in SLE(κ,ρ) processes to relate outer boundaries of hulls to dual SLE curves.

Experimental results

Research questions

• RQ1 How does Schramm-Loewner Evolution (SLE) arise as the scaling limit of discrete random curves such as loop-erased random walks and critical percolation interfaces?
• RQ2 What are the geometric and probabilistic properties of SLE for specific values of the parameter κ, such as κ=6 (locality) and κ=8/3 (restriction)?
• RQ3 Can critical exponents for planar Brownian motion, such as the disconnection exponent, be rigorously computed using SLE methods?
• RQ4 Is the time-reversal of a chordal SLEₖ curve distributed as another SLEₖ curve (reversibility), and for which κ values does this hold?
• RQ5 What is the relationship between SLE and conformal field theory, particularly through the KPZ formula and quantum gravity on random lattices?

Key findings

• The disconnection exponent for planar Brownian motion, i.e., the decay rate of the probability that a Brownian path disconnects the origin from infinity, is exactly 1/8, computed via SLE₆ and radial SLE₆.
• SLE₆ is the scaling limit of critical percolation interfaces on the triangular lattice, and its locality property is consistent with conformal invariance.
• SLE₈⁄₃ is the unique simple random curve satisfying the restriction property, and it describes the outer boundary of SLE₆ hulls.
• For κ=6, radial and chordal SLE are closely related, with radial SLE₆ providing a path to compute Brownian motion exponents.
• The conjectured duality between SLE(κ′,ρ) and SLE(16/κ′,ρ′) suggests a deep symmetry in outer boundaries of SLE hulls, though it remains unproven for general κ.
• Reversibility of SLEₖ holds for κ=2,6,8 and is conjectured for κ≤8, but fails for κ>8, indicating a phase transition in the process behavior.