Refined regularity of SLE
Yizheng Yuan
TL;DR
This work derives refined regularity statements for the SLE$_\kappa$ trace under capacity parametrisation by analyzing the forward Loewner equation. The authors establish sharp variation and Hölder-type moduli, with explicit exponents tied to the known optimal $p$-variation and Hölder exponents, and treat the special case $\kappa=8$ with additional techniques. They also obtain uniform-in-time estimates for the conformal maps that uniformise the evolving slitted domain for $\kappa\ge 8$, using a forward-flow/grid framework and the conformal-radius parametrisation. The results deepen understanding of the fine regularity of SLE traces and their associated conformal maps, with potential implications for Minkowski content, natural parametrisations, and boundary behaviour. Overall, the paper provides a comprehensive forward-Loewner-based approach to quantify sharp regularity scales for SLE across the noncritical and critical regimes, including uniform map control in the dense regime $\kappa\ge 8$.
Abstract
We prove refined (variation and Hölder-type) regularity statements for the SLE trace (under capacity parametrisation). More precisely, we show that the trace has finite $ψ$-variation for $ψ(x) = x^d(\log 1/x)^{-d-\varepsilon}$ and Hölder-type modulus $\varphi(t) = t^α(\log 1/t)^β$ where $d$ and $α$ are the optimal $p$-variation and Hölder exponents of SLE$_κ$ 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 $κ\ge 8$, we prove regularity estimates for the uniformising maps that hold uniformly in time, namely $\sup_t |\hat f_t'(u+iv)| \lesssim v^{2α-1}(\log 1/v)^β$ in case $κ>8$ and $v^{-1}(\log 1/v)^{-1/4}(\log\log 1/v)^{1+\varepsilon}$ in case $κ=8$. Our results are obtained from analysing the forward Loewner differential equation (in contrast to the other mentioned works which analyse the backward equation).