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October 2020 Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model
Janne Junnila, Eero Saksman, Christian Webb
Ann. Appl. Probab. 30(5): 2099-2164 (October 2020). DOI: 10.1214/19-AAP1553

Abstract

In this article we study imaginary Gaussian multiplicative chaos—namely a family of random generalized functions which can formally be written as $e^{iX(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbf{R}^{d}$, that is, it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions these objects live in, along with their basic stochastic properties, such as moment and tail estimates.

After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, that is, the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution.

The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

Citation

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Janne Junnila. Eero Saksman. Christian Webb. "Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model." Ann. Appl. Probab. 30 (5) 2099 - 2164, October 2020. https://doi.org/10.1214/19-AAP1553

Information

Received: 1 December 2018; Revised: 1 October 2019; Published: October 2020
First available in Project Euclid: 15 September 2020

MathSciNet: MR4149524
Digital Object Identifier: 10.1214/19-AAP1553

Subjects:
Primary: 60G20
Secondary: 82B20

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 5 • October 2020
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