Multipoint Estimates for Radial and Whole-Plane SLE

Abstract

We prove upper bounds for the probability that a radial SLEκ curve, κ ∈ (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 SLEκ curve. We then use these estimates to show that the lower Minkowski content of both the radial and whole-plane SLEκ traces restricted in a bounded region have finite moments of any order.

SLE as a Mating of Trees in Euclidean Geometry

“…We define the measure m η as a limit, rather than considering m(U ∩η([0, ∞))), since it is not known that the Minkowski content of η, which is a whole plane SLE κ (2 − κ), is well-defined near the origin. This assertion holds for whole plane SLE κ (see [LV17, Lemma 3.1] and [MZ17]), which means for κ = 2 we can remove the cutoff in (8).…”