[Paper Review] SLE coordinate changes

Oded Schramm, David B. Wilson|arXiv (Cornell University)|May 17, 2005

Stochastic processes and statistical mechanicsMathematics18 references126 citations

TL;DR

This paper unifies radial, chordal, and dipolar SLE by extending the SLE(κ;ρ) process to include interior force points, showing that radial SLE(κ) transforms into chordal SLE(κ;ρ) with ρ=κ−6 and vice versa via Möbius coordinate changes. It derives explicit martingales for the Radon–Nikodym derivative of SLE(κ;ρ₁,…,ρₙ) relative to standard SLE(κ), enabling exact likelihood ratios and applications to critical percolation and uniform spanning trees.

ABSTRACT

The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(kappa;rho) is extended to the setting where the force points can be in the interior of the domain, radial SLE(kappa) becomes chordal SLE(kappa;rho), with rho=kappa-6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(kappa;rho_1,...,rho_n) with respect to SLE(kappa).

Motivation & Objective

• To unify radial, chordal, and dipolar SLE into a single framework using coordinate transformations and extended SLE(κ;ρ) processes with interior force points.
• To establish that radial SLE(κ) transforms into chordal SLE(κ;ρ) with ρ=κ−6 under Möbius maps, and vice versa.
• To derive explicit martingales describing the Radon–Nikodym derivative of SLE(κ;ρ₁,…,ρₙ) with respect to standard SLE(κ), enabling likelihood ratio computations.
• To apply the derived martingales to estimate rare event probabilities in critical percolation and uniform spanning trees.

Proposed method

• Extends the SLE(κ;ρ) process to allow force points in the interior of the domain by defining a system of SDEs for the driving function Wₜ and force points Vⱼₜ.
• Uses Möbius transformations to relate radial and chordal SLE, showing that the radial SLE with drift corresponds to chordal SLE(κ;ρ) with ρ=κ−6.
• Derives the Radon–Nikodym derivative as a product of conformal derivatives and distance terms raised to power functions of ρⱼ and κ.
• Applies Itô's formula to the logarithm of the martingale to derive the SDE for the likelihood ratio, yielding explicit expressions in terms of gₜ′(z) and |zₜʲ−zₜᵏ|.
• Uses conformal invariance and scaling to relate discrete model probabilities to the martingale expressions in the continuum limit.
• Validates the martingale expressions by matching known exponents in critical percolation (κ=6) and uniform spanning trees (κ=2).

Experimental results

Research questions

• RQ1 How can radial, chordal, and dipolar SLE be unified under a single framework using coordinate transformations?
• RQ2 What is the precise form of the SLE(κ;ρ) process when force points are located in the interior of the domain rather than on the boundary?
• RQ3 How does the Radon–Nikodym derivative of SLE(κ;ρ₁,…,ρₙ) with respect to standard SLE(κ) depend on the conformal derivatives and mutual distances of force points?
• RQ4 What are the implications of the derived martingales for estimating the probability of rare events in critical statistical mechanics models?

Key findings

• Radial SLE(κ) transforms into chordal SLE(κ;ρ) with ρ=κ−6 under Möbius maps, and vice versa, establishing a direct duality between radial and chordal SLE with specific ρ values.
• The Radon–Nikodym derivative of SLE(κ;ρ₁,…,ρₙ) with respect to SLE(κ) is given by a product of terms involving |gₜ′(zⱼ)|^{ρⱼ²/(8κ)} and |zₜʲ−zₜᵏ|^{ρⱼρₖ/(4κ)} for j<k, with additional terms for interior force points.
• When κ=6 and each ρⱼ=2, the martingale simplifies to Mₜ = gₜ′(0)^{(n²−1)/12} ∏_{j<k} |zₜʲ−zₜᵏ|^{1/3}, matching the known exponent in critical percolation.
• For κ=2 and ρⱼ=2, the martingale becomes Mₜ = gₜ′(0)^{(n²−1)/4} ∏_{j<n} gₜ′(zⱼ) ∏_{j<k} |zₜʲ−zₜᵏ|, consistent with the exponent (n²−1)/4 in uniform spanning tree n-tuple points.
• The derived martingales are conformally invariant up to scaling by the conformal radius and provide a precise framework for computing rare event probabilities in discrete models.
• The framework extends the weighted SLE approach to include interior force points, enabling exact likelihood ratio computations in complex SLE configurations.