Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos
Brownian Loops, Layering Fields and Imaginary Gaussian Multiplicative Chaos
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DOI：
10.1007/s00220-020-03932-9
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发表时间：
2021-02-06
影响因子：
2.4
通讯作者：
Reddy, Tulasi Ram
中科院分区：
文献类型：
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作者：
Camia, Federico;Gandolfi, Alberto;Reddy, Tulasi Ram
我们研究领域让人想起顶点运营商建立从布朗回路汤在极限的回路汤强度趋于无穷大。更准确地说，按照Camia等人（Nucl Phys B 902：483-507，2016），我们在平面域中取一个（无质量或有质量的）布朗环汤，并为每个环分配一个随机符号。然后，我们考虑随机场定义，在域的每一点上，取纯虚常数的指数乘以与缠绕在该点上的环相关的符号之和。对于域共形相当于一个磁盘，和发散几何由于小循环，但我们表明，一个合适的重整化过程允许在适当的Sobolev空间中定义的字段。随后，我们让环汤的强度趋于无穷大，证明了这些顶点类场趋于共形协变随机场，该随机场可以表示为虚高斯乘性混沌的显式泛函，其协方差核由布朗环测度给出.除了利用布朗环汤和布朗环测度的性质外，我们分析的一个主要工具是类顶点场的线性泛函的显式Wiener-Ito混沌展开。我们的方法适用于模型的其他变体，例如，布朗环被磁盘取代。
We study fields reminiscent of vertex operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia et al. (Nucl Phys B 902:483-507, 2016), we take a (massless or massive) Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. For domains conformally equivalent to a disk, the sum diverges logarithmically due to the small loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener-Ito chaos expansion of linear functionals of vertex-like fields. Our methods apply to other variants of the model in which, for example, Brownian loops are replaced by disks.