Multipoint Estimates for Radial and Whole-plane SLE

Abstract

We prove upper bounds for the probability that a radial SLE$_{\kappa}$ curve, $\kappa\in(0,8)$, comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for a whole-plane SLE$_{\kappa}$ curve. We then use these estimates to show that the lower Minkowski content of both the radial and whole-plane SLE$_{\kappa}$ traces restricted in a bounded region have finite moments of any order.