Conformally invariant scaling limits: an overview and a collection of problems
Conformally invariant scaling limits: an overview and a collection of problems
复制标题
共形不变的缩放限制：概述和问题集合
DOI：
10.1007/978-1-4419-9675-6_34
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
O. Schramm
中科院分区：
文献类型：
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作者：
O. Schramm
二维统计物理学的许多数学模型要么是已知的，
或者被证明具有共形不变性。多年来，物理学家提出了预测
描述这些模型行为的各种指数。只是最近才有一些
这些预测可以用数学证明。新的发展之一是，
发现了一个单参数的随机曲线族，称为随机Loewner演化或
SLE。SLE曲线表现为在各种统计特性中出现的界面或路径的极限。
当在其上定义模型的网格的网格趋向于零时，物理模型的网格趋于零。
这篇文章的主要目的是列出一系列未决问题。一些
开放性问题表示物理知识中尚未被理解的方面
数学上。其他问题是关于SLE曲线本身的性质的问题。
在我们提出开放性问题之前，我们将对SLE的定义进行动机和解释，
并简要介绍了最近的成果。
Many mathematical models of statistical physics in two dimensions are either known
or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions
of various exponents describing the behavior of these models. Only recently have some of
these predictions become accessible to mathematical proof. One of the new developments is the
discovery of a one-parameter family of random curves called stochastic Loewner evolution or
SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical
physics models as the mesh of the grid on which the model is defined tends to zero.
The main purpose of this article is to list a collection of open problems. Some of the
open problems indicate aspects of the physics knowledge that have not yet been understood
mathematically. Other problems are questions about the nature of the SLE curves themselves.
Before we present the open problems, the definition of SLE will be motivated and explained,
and a brief sketch of recent results will be presented.