Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 203 / 4 / b / Solution

The scale is precisely the inverse conformal map:By definition, a chordal SLE in a specified scale in $(\mathbb{H},0,1)$ is a curve whose image under $\sigma$ is the half-plane-capacity-parameterized SLE from zero to infinity. Hereand the right-hand side is the given ${\mathrm{SLE}}_6$. Equivalently, this is Conformal invariance of SLE, with the target point transported from infinity to $1$. Thus $\tilde{\gamma }$ is ${\mathrm{SLE}}_6$ in $(\mathbb{H},0,1)$ in scale $\sigma$, using the original parameter $t$. This scaled parameter is distinguished from the ordinary half-plane-capacity parameterization of its image hulls used next.