Reconstructing the base field from imaginary multiplicative chaos

Abstract

We show that the imaginary multiplicative chaos $\exp (i\beta \Gamma )$ determines the gradient of the underlying field $\Gamma$ for all log-correlated Gaussian fields with covariance of the form $-\log |x-y|+g(x,y)$ with mild regularity conditions on $g$, for all $d\ge 2$ and for all $\beta \in (0,\sqrt{d})$. In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable w.r.t. its imaginary chaos.

Most notable changes in the title, unfortunately still no figures; but comments are always very welcome!