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 $bx\le\pi$.