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 $γ$ connecting $e^{ix}$ to 1 the probability of the event that $γ$ avoids the disk of radius $q$ centered at zero decays like $\exp(-bπx/(1-q))$ if either $b\in[5/8,1]\cup[5/4,\infty)$ and $x\in(0,π]$, or if $b\in(1,5/4)$, $x\in(0,π)$, and $bx\leπ$.