Oded Schramm: From Circle Packing to SLE

Regularity of SLE in $(t,κ)$ and refined GRR estimates

Schramm-Loewner evolution (SLE$_κ$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrtκ$ times Brownian motion. This yields a (half-plane) valued random field $γ= γ(t, κ; ω)$. (Hölder) regularity of in $γ(\cdot,κ;ω$), a.k.a. SLE trace, has been considered by many authors, starting with Rohde-Schramm (2005). Subsequently, Johansson Viklund, Rohde, and Wong (2014) showed a.s. Hölder continuity of this random field for $κ< 8(2-\sqrt{3})$. In this paper, we improve their result to joint Hölder continuity up to $κ< 8/3$. Moreover, we show that th…

Multiple backward Schramm--Loewner evolution and coupling with Gaussian free field

It is known that a backward Schramm--Loewner evolution (SLE) is coupled with a free boundary Gaussian free field (GFF) with boundary perturbation to give conformal welding of quantum surfaces. Motivated by a generalization of conformal welding for quantum surfaces with multiple marked boundary points, we propose a notion of multiple backward SLE. To this aim, we investigate the commutation relation between two backward Loewner chains, and consequently, we find that the driving process of each backward Loewner chain has to have a drift term given by logarithmic derivative of a partition functio…

Conformal removability of non-simple Schramm-Loewner evolutions

We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $κ\in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_κ$ is a.s. connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_1,\ldots,U_n$ with $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We show that the range of an SLE$_κ$ for $κ\in {\mathcal K}$ is a.s…

当 $κ$ 接近 8 时，Schramm-Loewner evolution 包含一个拓扑 Sierpiński carpet

Schramm-Loewner evolution contains a topological Sierpiński carpet when $κ$ is close to 8

本文证明了当 $κ$ 接近 8 时，SLE$_κ$ 曲线的值域几乎必然包含拓扑 Sierpiński carpet，从而得出 SLE$_κ$ 共形不可移除且共形焊接问题无唯一解，并证明其互补连通分量的邻接图不连通。

SLE曲线的尺寸

The dimension of the SLE curves

令$γ$为生成Schramm-Loewner演化（SLE）过程的曲线，参数为$κ\geq0$。我们证明，概率为1，豪斯多夫维数$γ$等于$\operatorname {Min}(2,1+κ/8)$。

Off-critical lattice models and massive SLEs

We suggest how versions of Schramm's SLE can be used to describe the scaling limit of some off-critical 2D lattice models. Many open questions remain.

SLE的正则性：至常数变化和连续模

Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity

本文研究了SLE的正则性，确定了至常数变化和连续模的最优界，并证明了自然参数化由ψ-变化的细网格极限给出。

Conformal welding problem, flow line problem, and multiple Schramm--Loewner evolution

A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with $N+1$ marked boundary points (MBPs) with $N\in\mathbb{Z}_{\ge 0}$. We propose the conformal welding problem for it in the case of $N\in\mathbb{Z}_{\ge 1}$. If $N=1$, it is reduced to the problem introduced by Sheffield, who solved it by coupling the QS with the Schramm--Loewner evolution (SLE). When $N \ge 3$, there natu…

Convergence of an algorithm simulating Loewner curves

The development of Schramm--Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde \cite{MR05}, is to sample Brownian motion at discrete times, interpolate appropriately in between and solve explicitly the Loewner equation with this approximation. This algorithm always produces piecewise smooth non self-intersecting curves whereas SLE$_κ$ has been proven to be simple for $κ\in[0,4]$, self-touching for $κ\in(4,8)$ and space-filling for $κ\geq 8$. In this p…

On Harmonic Measure of the Whole Plane Levy-Loewner Evolution

Levy-Loewner evolution (LLE) is a generalization of the Schramm-Loewner evolution (SLE) where the branching is possible in a course of growth process. We consider a class of radial Levy-Loewner evolutions for which sets of points of the average means beta-spectrum can be found exactly. In this paper we show how to overcome difficulties arised in previous works on multi-fractal analysis of SLE/LLE.