SLE coordinate changes

Stochastic Loewner evolution in doubly connected domains

This paper introduces the annulus SLE$_κ$ processes in doubly connected domains. Annulus SLE$_6$ has the same law as stopped radial SLE$_6$, up to a time-change. For $κ\not=6$, some weak equivalence relation exists between annulus SLE$_κ$ and radial SLE$_κ$. Annulus SLE$_2$ is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE$_2$ satisfies the reversibility property. We also consider the disc SLE$_κ$ process defined as a limiting case of the annulus SLE's. Disc SLE$_6$ has the same law as stopped full plane SLE$_6$, up to a tim…

On the Continuity of SLE(κ) in κ

We prove that for almost every Brownian motion sample, the corresponding SLE(κ) curves parameterized by capacity exist and change continuously in the supremum norm when κvaries in the interval [0,κ_0), where κ_0=8(2-\sqrt{3})=2.143... We estimate the κ-dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with κ.

Duality of Chordal SLE

We derive some geometric properties of chordal SLE$(κ;\vecρ)$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(κ;\vecρ)$ process has the same distribution as the image of a chordal SLE$(κ';\vec{ρ'})$ trace, where $κ>4$, $κ'=16/κ$, and the forces $\vecρ$ and $\vec{ρ'}$ are suitably chosen. We find that for $κ\ge 8$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from $x$. Then we obtain a new proof…

多弦 SLE 的交换关系分类

Classification of commutation relation for multi-chordal SLE

本文对多弦 SLE 的 BPZ 方程线性解空间进行分类，通过三角坐标变换化为有限维线性代数，确定了平移不变解空间的维数，恢复了 Catalan 维数公式，并构造了高斯自由场水平线的配分函数。

Schramm-Loewner Evolution

This is the first expository set of notes on SLE I have written since publishing a book two years ago [45]. That book covers material from a year-long class, so I cannot cover everything there. However, these notes are not just a subset of those notes, because there is a slight change of perspective. The main differences are: o I have defined SLE as a finite measure on paths that is not necessarily a probability measure. This seems more natural from the perspective of limits of lattice systems and seems to be more useful when extending SLE to non-simply connected domains. (However, I do not di…

Coordinates changed random fields on the sphere

We construct time dependent random fields on the sphere through coordinates change and subordination and we study the associated angular power spectrum. Some of this random fields arise naturally as solutions of partial differential equations with random initial condition represented by a Gaussian random field.