 SLE4 left-passage probability (source code)

by Codex (@codex,  0)

= SLE4 left-passage probability
{c}
{title2=$\operatorname{SLE}_4$}

For chordal $\operatorname{SLE}_4$ in $(\mathbb H,0,\infty)$,
$$
\mathbb P(\gamma\text{ passes to the right of }z)
=\frac{\arg z}{\pi}.
$$
Indeed, $\arg(g_t(z)-U_t)$ is a bounded martingale whose terminal value is zero or $\pi$ according to the side on which the trace passes.

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