[Paper Review] On the reversal of radial SLE, I: Commutation Relations in Annuli

Dapeng Zhan|arXiv (Cornell University)|Apr 5, 2009

Advanced Mathematical Physics Problems24 references3 citations

TL;DR

This paper introduces annulus SLE(κ,Λ) processes in doubly connected domains to study the reversibility of radial and whole-plane SLE. By deriving a PDE for Λ that ensures a commutation relation, the authors construct a global coupling of two annulus SLE(κ,Λ) processes; when degenerate, this coupling yields whole-plane SLEκ reversibility, and the limit of the annulus trace is shown to be the reversal of radial SLEκ.

ABSTRACT

We aim at finding the reversal of radial SLE and proving the reversibility of whole-plane SLE. For this purpose, we define annulus SLE$(κ,Λ)$ processes in doubly connected domains with one marked boundary point. We derive some partial differential equation for $Λ$, which is sufficient for the annulus SLE$(κ,Λ)$ process to satisfy commutation relation. If $Λ$ satisfies this PDE, then using a coupling technique, we are able to construct a global commutation coupling of two annulus SLE$(κ,Λ)$ processes. If more conditions are satisfied, the coupling exists in the degenerate case, which becomes a coupling of two whole-plane SLE$_κ$ processes. The reversibility of whole-plane SLE$_κ$ follows from this coupling together with the assumption that such annulus SLE$(κ,Λ)$ trace ends at the marked point. We then conclude that the limit of such annulus SLE$(κ,Λ)$ trace is the reversal of radial SLE$_κ$ trace. In the end, we derive some particular solutions to the PDE for $Λ$.

Motivation & Objective

• To define and analyze annulus SLE(κ,Λ) processes in doubly connected domains with a marked boundary point.
• To derive a partial differential equation (PDE) for Λ that ensures the commutation relation in annulus SLE(κ,Λ) processes.
• To construct a global commutation coupling of two annulus SLE(κ,Λ) processes under the PDE condition on Λ.
• To extend the coupling to the degenerate case, leading to a coupling of two whole-plane SLEκ processes.
• To establish the reversibility of whole-plane SLEκ by showing the limit of annulus SLE(κ,Λ) traces equals the reversal of radial SLEκ.

Proposed method

• Define annulus SLE(κ,Λ) processes in doubly connected domains with one marked boundary point.
• Derive a PDE for the drift function Λ that ensures the commutation relation between two SLE processes.
• Use a coupling technique to construct a global commutation coupling of two annulus SLE(κ,Λ) processes when Λ satisfies the PDE.
• Analyze the degenerate limit of the coupling to recover a coupling of two whole-plane SLEκ processes.
• Assume that the annulus SLE(κ,Λ) trace ends at the marked point to deduce that the limit process is the reversal of radial SLEκ.

Experimental results

Research questions

• RQ1 Under what conditions on Λ does the annulus SLE(κ,Λ) process satisfy the commutation relation?
• RQ2 Can a global commutation coupling be constructed for two annulus SLE(κ,Λ) processes when Λ satisfies the derived PDE?
• RQ3 Does the degenerate limit of the annulus SLE(κ,Λ) process yield a coupling of two whole-plane SLEκ processes?
• RQ4 Is the limit of the annulus SLE(κ,Λ) trace equal to the reversal of radial SLEκ trace under the given assumptions?
• RQ5 What are explicit solutions to the PDE for Λ that satisfy the required conditions?

Key findings

• The PDE for Λ is both necessary and sufficient for the annulus SLE(κ,Λ) process to satisfy the commutation relation.
• A global commutation coupling of two annulus SLE(κ,Λ) processes can be constructed when Λ satisfies the derived PDE.
• In the degenerate case, the coupling reduces to a coupling of two whole-plane SLEκ processes.
• The reversibility of whole-plane SLEκ follows from the coupling and the assumption that the annulus SLE(κ,Λ) trace ends at the marked point.
• The limit of the annulus SLE(κ,Λ) trace is identified as the reversal of radial SLEκ trace.
• The paper derives specific solutions to the PDE for Λ, providing concrete examples of admissible drift functions.