[论文解读] Brownian Loops and Conformal Fields

Federico Camia|arXiv (Cornell University)|Jan 20, 2015

Stochastic processes and statistical mechanics参考文献 39被引用 3

一句话总结

本文建立了平面随机游走环流在缩放极限下成为布朗运动环流的结果，即一个共形不变的连续环的泊松点过程。它从环流中构造出类似相关函数的量，其行为类似于共形场论（CFT）中初级场的相关函数，通过共形不变性与布朗运动环的测度论性质，建立了统计力学缩放极限与CFT之间的严格联系。

ABSTRACT

The main topic of these lecture notes is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical statistical physics is that scaling limits provide the link between statistical mechanics and (Euclidean) field theory. In order to explain the main ideas behind the concept of scaling limit, I will focus on a "toy" model that exhibits the typical behavior of statistical mechanical models at and near the critical point. This model, known as the random walk loop soup, is actually interesting in its own right. It can be described as a Poissonian ensemble of lattice loops, or a lattice gas of loops since it fits within the ideal gas framework of statistical mechanics. After introducing the model and discussing some interesting connections with the discrete Gaussian free field, I will present some results concerning its scaling limit, which leads to a Poissonian ensemble of continuum loops known as the Brownian loop soup. The latter was introduced by Lawler and Werner and is a very interesting object with connections to the Schramm-Loewner Evolution and various models of statistical mechanics. In the second part of the lectures, I will use the Brownian loop soup to construct a family of functions that behave like correlation functions of a conformal field. I will then use these functions and their derivation to introduce the concept of conformal field and to explore the connection between scaling limits and conformal fields.

研究动机与目标

• 建立平面随机游走环流在缩放极限下成为布朗运动环流的结果。
• 证明布朗运动环流满足共形限制，这是临界系统关键对称性质的体现。
• 从环流中构造出行为类似共形场论（CFT）中初级场相关函数的函数。
• 通过随机过程与共形不变性，提供一个严格的数学框架，将统计力学的缩放极限与CFT联系起来。
• 证明布朗运动环测度在乘法常数意义下的唯一性与缩放性质。

提出的方法

• 将随机游走环流用作格点环气体的离散模型，定义为格点环上的泊松点过程。
• 分析当格点网格趋于零时环流的缩放极限，证明其收敛于连续布朗运动环的泊松过程。
• 应用共形限制性质来刻画缩放极限测度，证明其在常数倍意义下唯一。
• 通过积分那些与某点相交或环绕该点的环，从环测度中推导出类似相关函数的量。
• 利用布朗运动的马尔可夫性质与缩放性，对环形区域中的环测度进行有界处理，并建立其与半径的对数缩放关系。
• 利用已知的关于填充布朗运动环期望面积的结果，计算环绕数测度，并推导出k重环绕环的精确对数缩放关系。

实验结果

研究问题

• RQ1 随机游走环流在缩放极限下是否收敛于一个共形不变的连续过程？
• RQ2 是否可以利用布朗运动环流构造出类似共形场论（CFT）中初级场相关函数的函数？
• RQ3 环测度在环形区域中的精确缩放行为是什么？其与共形不变性有何关系？
• RQ4 环围绕某一点的环绕数如何影响测度？其与径向尺度的精确依赖关系如何？
• RQ5 布朗运动环测度是否由其共形限制性质唯一确定？

主要发现

• 布朗运动环流是在平面中唯一的共形不变泊松点过程环，至多相差一个乘法常数。
• 包含于内径δ与外径R的环形区域中且与某点相交的环的测度，按 (1/5) log(R/δ) 的方式对数缩放。
• 对于恰好围绕某点k圈的环，其测度按 (1/(2π²k²)) log(R/δ) 缩放，其中精确常数由已知面积结果推导得出。
• 围绕原点k圈的填充布朗运动环的期望面积为 1/(2πk²)，其中k ≠ 0。
• 环测度满足共形限制，且该性质在常数倍意义下唯一确定了该测度。
• 从环流中构造出类似CFT的相关函数，为二维临界系统中初级场相关性的随机过程实现提供了严格数学基础。