Multipoint Estimates for Radial and Whole-plane SLE

TL;DR

This paper establishes upper bounds on the probability of radial and whole-plane SLE curves approaching multiple points, and demonstrates that their Minkowski content has finite moments, advancing understanding of their geometric properties.

Contribution

It provides new probabilistic estimates for multiple point proximity in radial and whole-plane SLE, and proves finiteness of moments of their Minkowski content.

Findings

Upper bounds for multi-point proximity probabilities in radial SLE.

Upper bounds for multi-point proximity probabilities in whole-plane SLE.

Finite moments of Minkowski content for SLE traces in bounded regions.

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.