# LA Probability Forum

*,*Department of Mathematical Sciences

*,*Tsinghua University

*,*

*UCLA, Math Scienes Bldg., Rm 6627*

In 1999, O. Schramm introduced Schramm—Loewner evolution (SLE) as a non-self-crossing random curve driven by a multiple of Brownian motion using Loewner's transform. This definition is motivated by a quest to describe the random interfaces in 2D critical lattice models, which satisfy the conformal invariance and domain Markov property. In this talk, we will consider the law of a pair of two SLEs with conformal invariance and domain Markov property, following the framework of Dub'edat's commutation relation.

Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE for $\kappa\le 4$: it is either a two-sided radial SLE with spiral of constant spiraling rate or a chordal SLE weighted by a power of the conformal radius of its complement. Two-sided radial SLE with spiral is a generalization of two-sided radial SLE (without spiral) and satisfies the resampling property. However, unlike in the chordal case, the resampling property does not uniquely determine the pair due to the additional degree of freedom in the spiraling rate. We also discuss the semiclassical limit of the commutation relation as $\kappa\to 0$.

This talk is based on a joint work with Yilin Wang (IHES, France).