Reconstructing the base field from imaginary multiplicative chaos

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