An application of Jacobi's elliptic functions to asymptotic probabilities for conformal restriction measures

Abstract

We show that for the conformal restriction measure with exponent $b$ in the unit disk on hulls $\gamma$ connecting $e^{ix}$ to 1 the probability of the event that $\gamma$ avoids the disk of radius $q$ centered at zero decays like $\exp (-b\pi x/(1-q))$ if either $b\in [5/8,1]\cup [5/4,\infty )$ and $x\in (0,\pi ]$, or if $b\in (1,5/4)$, $x\in (0,\pi )$, and $bx\le \pi$.

9 pages