双环Loewner势

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 rate of convergence of loop-erased random walk to SLE(2)

We derive a rate of convergence of the Loewner driving function for planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). The proof uses a new estimate of the difference between the discrete and continuous Green's functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE(2), we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.

Quantum Loewner Evolution

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model} $η$-DBM, a generalization of DLA in which particle locations are sampled from the $η$-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider $η$-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter $γ\in [0…

共形反常的普适性

Universality of the conformal anomaly

该研究证明局部实一维模函子的普适性质，将其与普适Liouville作用关联，为共形场论和Schramm-Loewner演化中中心荷的出现提供数学解释，还证明布朗环测度作为SLE典范限制函数的唯一性。

$\text{SLE}_κ$ 环路测度的 Onsager–Machlup 泛函

Onsager-Machlup functional for $\text{SLE}_κ$ loop measures

本文建立 Brown 环路测度两种重整化的联系，并据此将 Loewner 能量表述为 $\text{SLE}_κ$ 及同中心荷 MKS 环路测度的 Onsager–Machlup 泛函。

关于焊接映射的Loewner能量

On the Loewner energy of a welding homeomorphism

本文研究焊接映射的Loewner能量，引入操作符Λ_φ并证明其与单层势和复合算子的关系，推导出Loewner能量的表达式及Weil-Petersson条件。

共形缝合与物质--Liouville--鬼场因子化

Conformal welding and the matter--Liouville--ghost factorisation

本文研究局部共形变换在GFF和SLE相关测度上的作用，主要创新是为SLE loop measure的缝合同胚提出Cameron--Martin型公式，并引入量子曲面共形缝合的新方法。

SLE焊接测度的拟不变性

Quasi-invariance for SLE welding measures

本文研究SLE焊接测度的性质，证明其在Weil-Petersson圆周同胚的群作用下具有拟不变性，为理解SLE焊接的分布规律提供了新的理论结果。