Conformally Invariant Processes in the Plane
Conformally Invariant Processes in the Plane
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DOI：
10.1090/surv/114
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发表时间：
2005
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影响因子：
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通讯作者：
G. Lawler
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文献类型：
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作者：
G. Lawler
理论物理学家预测，统计物理中许多二维晶格模型的标度极限在某种意义上是共形不变的。这一信念使物理学家能够预测这些关键系统的许多量。这些标度极限的本质最近已经通过一个著名的工具布朗运动和一个新的构造Schramm-Loewner演化(SLE)得到了精确的描述。这本书是对共形不变过程的介绍，这些过程看起来像是比例极限。内容包括：随机积分；布朗运动衍生的复布朗运动和测度；共形映射和单叶函数；Loewner微分方程和Loewner链；Schramm-Loewner演化(SLE)，即具有布朗运动输入的Loewner链；以及布朗运动的相交指数的应用。必修课是研究生一年级的实数分析、复数分析和概率论课程。本书适合对随机过程及其在理论物理中的应用感兴趣的研究生和研究数学家。
Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.