Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 203 / 2 / c

For $z=x+iy$ with $|x|\le 1$ and $0<y\le 1$, the given Reverse SLE derivative martingale starts fromIt is a nonnegative local martingale and therefore a supermartingale. Since its second factor is at least one,

Choosewhich is possible exactly because $\kappa >8$. At height $y_n=2^{-n}$, take a horizontal grid of spacing comparable to $y_n$ in $[-1,1]$. The Markov inequality gives, at each grid point,There are $O(y_n^{-1})$ grid points, so the probability that the bound fails anywhere on level $n$ is at most $C{y}_n^{1-2\alpha -8/\kappa }$. These probabilities are summable. The Borel-Cantelli lemmas therefore give an almost surely finite random constant controlling every sufficiently fine grid, and enlarging it handles the finitely many remaining levels.

Every point of the half-rectangle lies within a fixed hyperbolic distance of one of these grid points at comparable height. The Koebe distortion theorem compares the two derivatives by a universal factor. Hence an almost surely finite random $C$ satisfiesfor all $x\in [-1,1]$ and $0<y\le 1$. This proves the Reverse SLE derivative bound above the space-filling threshold.