[Paper Review] Imaginary geometry III: reversibility of SLE_κ for κ\in (4,8)

Jason Miller, Scott Sheffield⋆|DSpace@MIT (Massachusetts Institute of Technology)|Jan 6, 2012

Mathematical Dynamics and FractalsMathematics29 references69 citations

TL;DR

This paper establishes the time-reversal symmetry of chordal SLE$_\kappa$ processes for $\kappa \in (4,8)$, proving that the time-reversal of an SLE$_\kappa$ curve from $x$ to $y$ in a Jordan domain $D$ is distributed as an SLE$_\kappa$ curve from $y$ to $x$, up to reparameterization. The result extends to $\mathrm{SLE}_\kappa(\rho_1;\rho_2)$ processes when $\rho_1, \rho_2 \geq \kappa/2 - 4$, the critical threshold for boundary-filling behavior, and provides a key ingredient for defining canonical, continuous conformal loop ensembles $\mathrm{CLE}_\kappa$ and coupling Gaussian free fields with piecewise constant differences along $\mathrm{SLE}_\kappa$ boundaries.

ABSTRACT

Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix κ\in (4,8) and let η be an SLE_κprocess from x to y in D. We prove that the law of the time-reversal of ηis, up to reparameterization, an SLE_κprocess from y to x in D. More generally, we prove that SLE_κ(ρ_1;ρ_2) processes are reversible if and only if both ρ_i are at least κ/2-4, which is the critical threshold at or below which such curves are boundary filling. Our result supplies the missing ingredient needed to show that for all κ\in (4,8) the so-called conformal loop ensembles CLE_κ are canonically defined, with almost surely continuous loops. It also provides an interesting way to couple two Gaussian free fields (with different boundary conditions) so that their difference is piecewise constant and the boundaries between the constant regions are SLE_κcurves.

Motivation & Objective

• To establish the time-reversal symmetry of chordal $\mathrm{SLE}_\kappa$ processes for $\kappa \in (4,8)$, a regime where reversibility was previously unproven.
• To extend this result to $\mathrm{SLE}_\kappa(\rho_1;\rho_2)$ processes, identifying the critical threshold $\rho_i \geq \kappa/2 - 4$ for reversibility.
• To resolve a key open problem in conformal loop ensemble theory by proving that $\mathrm{CLE}_\kappa$ for $\kappa \in (4,8)$ consists of almost surely continuous loops, independent of the starting point of the branching $\mathrm{SLE}$ process.
• To construct a coupling of two Gaussian free fields with different boundary conditions such that their difference is piecewise constant, with discontinuities along $\mathrm{SLE}_\kappa$ curves.
• To provide a rigorous foundation for the duality and reversibility principles in imaginary geometry, particularly for $\kappa \in (4,8)$, using light cone and flow line techniques.

Proposed method

• The proof relies on the light cone characterization of $\mathrm{SLE}_{\kappa'}$ traces from [MS12a], which relates outer boundaries of $\mathrm{SLE}_{\kappa'}$ to $\mathrm{SLE}_\kappa$ processes via the Gaussian free field.
• It uses the continuity of $\mathrm{SLE}_\kappa(\underline{\rho})$ and $\mathrm{SLE}_{\kappa'}(\underline{\rho}')$ traces even when interacting non-trivially with the boundary, established in [MS12a].
• The authors reduce the general case to a critical case using a recursive iteration procedure that exhausts the curve, ensuring the time-reversal process remains within the $\mathrm{SLE}_{\kappa'}(\rho_1;\rho_2)$ class.
• They employ a coupling of two Gaussian free fields $h$ and $\widetilde{h}$ such that $h - \widetilde{h}$ is piecewise constant, with jumps along the $\mathrm{SLE}_{\kappa'}(\rho^L;\rho^R)$ path, and use the reversibility of the path to show the time-reversed process corresponds to the flow line of $\widetilde{h}$.
• The method leverages the coordinate change formula for conformal maps and the behavior of boundary conditions under rotation, particularly in the strip domain, to verify the symmetry of the path law.
• It builds on the commutativity and duality techniques from [Dub07, Sch00, Zha08, MS12b], but extends them to the non-simple, boundary-filling regime for $\kappa \in (4,8)$.

Experimental results

Research questions

• RQ1 Is the time-reversal of an $\mathrm{SLE}_\kappa$ process from $x$ to $y$ in a Jordan domain $D$ distributed as an $\mathrm{SLE}_\kappa$ process from $y$ to $x$, for $\kappa \in (4,8)$?
• RQ2 What is the precise condition on the $\rho$-parameters in $\mathrm{SLE}_\kappa(\rho_1;\rho_2)$ processes that ensures time-reversal symmetry?
• RQ3 Can the construction of $\mathrm{CLE}_\kappa$ for $\kappa \in (4,8)$ be made canonical, independent of the initial point of the branching $\mathrm{SLE}_{\kappa}(\kappa-6)$ process, given reversibility?
• RQ4 How can two Gaussian free fields with different boundary conditions be coupled so that their difference is piecewise constant and the discontinuity lines are $\mathrm{SLE}_\kappa$ curves?
• RQ5 What is the probabilistic interpretation of the probability $P_L(z)$ that a point $z$ lies to the left of a $\mathrm{SLE}_{\kappa'}(\rho^L;\rho^R)$ path, particularly when $\rho^L = \rho^R = \kappa'/2 - 4$?

Key findings

• The time-reversal of an $\mathrm{SLE}_\kappa$ process for $\kappa \in (4,8)$ is distributed as an $\mathrm{SLE}_\kappa$ process from $y$ to $x$, up to reparameterization, establishing full time-reversal symmetry in this regime.
• The $\mathrm{SLE}_\kappa(\rho_1;\rho_2)$ process is reversible if and only if $\rho_1, \rho_2 \geq \kappa/2 - 4$, which is the critical threshold below which such curves are boundary-filling.
• This result implies that $\mathrm{CLE}_\kappa$ for $\kappa \in (4,8)$ is canonically defined with almost surely continuous loops, independent of the starting point of the branching $\mathrm{SLE}$ process.
• The paper constructs a coupling of two Gaussian free fields $h$ and $\widetilde{h}$ such that $h - \widetilde{h}$ is piecewise constant, with the boundary between constant regions being an $\mathrm{SLE}_\kappa(\rho^L;\rho^R)$ curve, and the time-reversal of the path corresponds to the flow line of $\widetilde{h}$.
• For $\kappa' \in (4,8)$, the probability $P_L(z)$ that a point $z$ lies to the left of the $\mathrm{SLE}_{\kappa'}(\rho^L;\rho^R)$ path is a linear function that takes value $1 - \kappa'/4$ on the left boundary and $\kappa'/4$ on the right, with $\kappa'/4 \in (1/2,1)$, indicating a higher likelihood of being to the right of the path near the left boundary.
• When $\kappa' \to 8$, the path becomes space-filling and $P_L(z) \to 1/2$, reflecting the symmetric passage of the path around any given point.