Imaginary multiplicative chaos: Moments, regularity and connections to the Ising model

Understand

In this article we study imaginary Gaussian multiplicative chaos -- namely a family of random generalized functions which can formally be written as $e^{i X(x)}$, where $X$ is a log-correlated real-valued Gaussian field on $\mathbb{R}^d$, i.e.

• 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 do 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 so called critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, i.e.