[Paper Review] Two-sided radial SLE and length-biased chordal SLE

Laurence S. Field|arXiv (Cornell University)|Jan 13, 2016

Geometric Analysis and Curvature Flows18 references4 citations

TL;DR

This paper establishes that for $\kappa \leq 4$, the integral over all interior points $\zeta$ of two-sided radial SLE$_\kappa$ measures (rooted at $\zeta$) yields a measure equivalent to chordal SLE$_\kappa$ biased by the curve's natural length—specifically, its $(1 + \kappa/8)$-dimensional Minkowski content. This confirms a continuum analog of the discrete loop measure unrooting principle, extending conformal invariance and length-biasing duality to SLE processes.

ABSTRACT

We show that, for $κ\le 4$, the integral of the two-sided radial SLE$_κ$ measures over all interior points is chordal SLE$_κ$ biased by the path's natural length, which is its $(1+κ/8)$-dimensional Minkowski content.

Motivation & Objective

• To establish a continuum analog of the discrete unrooting principle for loop measures, where aggregating rooted measures yields a length-biased unrooted measure.
• To clarify the relationship between two-sided radial SLE$_\kappa$ (rooted at interior points) and chordal SLE$_\kappa$ (unrooted) in simply connected planar domains.
• To prove that the aggregate measure of two-sided radial SLE$_\kappa$ over all interior points is equivalent to chordal SLE$_\kappa$ biased by the curve's natural length, defined via its $(1 + \kappa/8)$-dimensional Minkowski content.
• To extend the framework of conformal invariance and path-length biasing from Brownian loop measures to SLE processes, particularly for $\kappa \leq 4$.

Proposed method

• Define two-sided radial SLE$_\kappa$ as chordal SLE$_\kappa$ conditioned to pass through an interior point $\zeta$, using the Green’s function $G_D(\zeta)$ as the normalization factor.
• Construct the aggregate measure $\nu = \int_D \mu_\zeta \, dA(\zeta)$, where $\mu_\zeta$ is the two-sided radial SLE$_\kappa$ measure rooted at $\zeta$, and $D$ is a simply connected domain with analytic boundary.
• Use the natural parametrization of SLE$_\kappa$ curves, defined via the $(1 + \kappa/8)$-dimensional Minkowski content, to express the length bias in the target measure.
• Apply a Riemann integral framework on the space of curves under a natural metric $d_S$ to define the integral of measures over $D$, ensuring convergence and well-definedness.
• Establish convergence via Prokhorov distance estimates by partitioning the domain into dyadic rectangles and comparing the rooted measure sum to the aggregate measure.
• Prove integrability of the Green’s function $G_D(\zeta)$ over $D$, ensuring the improper integral $\int_D G_D(\zeta) \, dA(\zeta)$ is finite and defines the length-biased chordal SLE$_\kappa$ measure.

Experimental results

Research questions

• RQ1 How does the integral of two-sided radial SLE$_\kappa$ measures over all interior points relate to the unrooted chordal SLE$_\kappa$ measure for $\kappa \leq 4$?
• RQ2 Can the unrooting principle from discrete loop measures—where summing over roots yields a length-biased unrooted measure—be extended to the continuum SLE setting?
• RQ3 What is the precise form of the Radon-Nikodym derivative between the aggregate of rooted measures and the chordal SLE$_\kappa$ measure?
• RQ4 Does the natural parametrization of SLE$_\kappa$ curves, via their $(1 + \kappa/8)$-dimensional Minkowski content, provide the correct length bias in the continuum limit?
• RQ5 Under what conditions on the domain $D$ (e.g., analytic vs. piecewise $C^1$ boundary) does the result hold, and how does the integrability of the Green’s function affect the construction?

Key findings

• For $\kappa \leq 4$, the aggregate measure $\nu = \int_D \mu_\zeta \, dA(\zeta)$ is equal in law to chordal SLE$_\kappa$ biased by the curve’s natural length, i.e., $\frac{d\nu}{d\mu}(\gamma) = |\gamma|$, where $|\gamma|$ is the $(1 + \kappa/8)$-dimensional Minkowski content of $\gamma$.
• The Green’s function $G_D(\zeta)$ for chordal SLE$_\kappa$ is integrable over $D$, ensuring the improper integral $\int_D G_D(\zeta) \, dA(\zeta)$ is finite and defines a well-behaved measure.
• The convergence of the Riemann sum approximation $\sum_{Q} \mu_{z_Q} A(Q)$ to the aggregate measure $\nu$ is controlled by a Prokhorov distance estimate of order $\epsilon^{\alpha/2}$, with $\alpha > 0$, confirming the integral’s existence.
• The result holds for unbounded domains $D$ with analytic boundary at infinity if and only if $\kappa < 8(\sqrt{2} - 1) \approx 3.3137$, due to integrability of $G_D$ at infinity.
• For domains with piecewise $C^1$ boundaries, the result still holds provided the area of the $\epsilon^{1/2}$-neighborhood of the boundary decays as a power of $\epsilon$, ensuring the discarded measure is negligible.
• The proof relies on conformal invariance, the domain Markov property, and reversibility of SLE$_\kappa$ for $\kappa \leq 4$, which are essential for constructing the rooted measures and analyzing their aggregate.