Brownian loop catchers with central charges $-2le c<0$
Gefei Cai, Peking University
We introduce a family of random closed subsets of Brownian traces, called Brownian loop catchers, with central charge in the range $-2le c<0$. Such a Brownian loop catcher with central charge $c$, together with the loops intersecting it from an independent Brownian loop soup of intensity $-c/2$, recovers the original Brownian trace in law. At $c=-2$, this object is the continuum loop-erased random walk, which is SLE$_2$ in two dimensions; as $cuparrow 0$, it degenerates to the Brownian trace itself. We prove that in two dimensions, the outer boundary of a Brownian loop catcher with central charge $-2le c<0$ is locally SLE$_kappa$, where $c=1-6left(frac{2}{sqrt{kappa}}-frac{sqrt{kappa}}{2}right)^2$. Consequently, a planar Brownian trace contains an SLE$_kappa$-type curve for every $kappain[2,8/3]$.