A Dimension Spectrum for SLE Boundary Collisions

Abstract

We consider chordal SLE(kappa) curves for kappa > 4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension min {2-8/kappa,1}. We study the random sets of points at which the curve collides with the real line at a specified "angle" and compute an almost sure dimension spectrum describing the metric size of these sets. We work with the forward SLE flow and a key tool in the analysis is Girsanov's theorem, which is used to study events on which moments concentrate. The two-point correlation estimates are proved using the direct method.Comment: 21 page

Integral Means Spectrum of Whole-Plane SLE

“…We conclude that F behaves like ψ (4.1), up to arbitrary small logarithmic correction. 1 This completes the proof of Theorem 1.2 for t < t 1 and t / ∈ T κ . Lastly, when t belongs to the discrete set T κ (3.17), because of (3.19), g 0 (u) (3.5) vanishes too fast at u = 0, and neither an upper bound like (4.3) holds, nor Lemma 4.2.…”

“…Before we proceed with the details of the analysis, let us also mention the work by Johansson Viklund and Lawler [11], who established the almost sure version of the SLE tip multifractal spectrum, that by Alberts, Binder and Viklund [1] on the almost sure dimension spectrum for SLE boundary collisions, as well as the recent preprint by Gwynne, Miller, and Sun [9], who used the so-called "Imaginary Geometry" of Miller and Sheffield [18,19,20,21] to compute the a.s. value of the SLE bulk multifractal spectrum.…”

Schramm’s Formula and the Green’s Function for Multiple SLE

“…decays deterministically; this is called the radial parametrization in this context. Here O t is defined as the image under g t of the rightmost point in the hull at time t; in particular, O t = g t (0+) if 0 < κ 4, see, e.g., [1]. Geometrically C t equals (1/4 times) the conformal radius seen from ξ 2 in H t after Schwarz reflection.…”

“…The key observation is that under the measure P * (corresponding toρ) we have that s(t) < ∞ almost surely and that t is positive recurrent and converges to an invariant distribution. This uses ρ > κ/2 − 4 so thatρ < κ/2 − 4; see, e.g., [1] and [25,Section 5.2]. In fact, we have the following formula for the limiting distribution (set ν = −r κ (κ − 8 − ρ) = β + aρ/2 in Lemma 2.2 of [1]):…”

“…This uses ρ > κ/2 − 4 so thatρ < κ/2 − 4; see, e.g., [1] and [25,Section 5.2]. In fact, we have the following formula for the limiting distribution (set ν = −r κ (κ − 8 − ρ) = β + aρ/2 in Lemma 2.2 of [1]):…”