Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 3 / c / ii

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- Solution ii

Solution

The imaginary part

ar g (g_{t} (z) - U_{t})

is a bounded local martingale and hence a martingale. As the simple transient trace passes

z

, this angle converges to

π

if the trace passes to the right of

z

and to

0

if it passes to the left. Bounded convergence therefore gives

ar g z = π P (γ passes to the right of z),

so the SLE4 left-passage probability is

P (γ passes to the right of z) = \frac{ar g z}{π} .

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