Refined regularity of SLE

TL;DR

This paper establishes refined regularity properties of SLE traces, including variation and H"older continuity, and provides uniform estimates for the associated conformal maps, advancing understanding of their fine geometric structure.

Contribution

It proves new regularity results for SLE traces and uniformising maps, including optimal variation and H"older exponents, using analysis of the forward Loewner differential equation.

Findings

SLE trace has finite a-variation with a(x) = x^d(\,log 1/x)^{-d-\u03b5}

Hf6lder-type modulus a(t) = t^(\,log 1/t)^{} with optimal exponents

Uniform bounds on f_t' for b  8, including explicit logarithmic factors

Abstract

We prove refined (variation and H\"older-type) regularity statements for the SLE trace (under capacity parametrisation). More precisely, we show that the trace has finite $\psi$-variation for $\psi(x) = x^d(\log 1/x)^{-d-\varepsilon}$ and H\"older-type modulus $\varphi(t) = t^\alpha(\log 1/t)^{\beta}$ where $d$ and $\alpha$ are the optimal $p$-variation and H\"older exponents of SLE$_\kappa$ which have been previously identified by Viklund, Lawler (2011) and Friz, Tran (2017). For SLE$_8$, we simplify a step in the proof by Kavvadias, Miller, and Schoug (2021), and get the modulus $\varphi(t) = (\log 1/t)^{-1/4}(\log\log 1/t)^{2+\varepsilon}$. Finally, for $\kappa \ge 8$, we prove regularity estimates for the uniformising maps that hold uniformly in time, namely $\sup_t |\hat f_t'(u+iv)| \lesssim v^{2\alpha-1}(\log 1/v)^\beta$ in case $\kappa>8$ and $v^{-1}(\log 1/v)^{-1/4}(\log\log 1/v)^{1+\varepsilon}$ in case $\kappa=8$. Our results are obtained from analysing the forward Loewner differential equation (in contrast to the other mentioned works which analyse the backward equation).