Abstract
AbstractWe 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–Itô 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.
Funder
Université du Luxembourg
New York University Abu Dhabi
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference66 articles.
1. Aizenman, M.: Geometric analysis of $$\phi ^4$$ fields and Ising models. Parts I and II. Commun. Math. Phys. 86, 1–48 (1982)
2. Astala, K., Jones, P., Kupiainen, A., Saksman, E.: Random conformal weldings. Acta Math. 207(2), 203–254 (2011)
3. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241(2), 333–380 (1984)
4. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry of critical fluctuations in two dimensions. J. Stat. Phys. 34(5–6), 763–774 (1984)
5. Berestycki, N.: An elementary approach to Gaussian multiplicative chaos. Electron. Commun. Probab. 22(27), 12 (2017)
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献