Reconstructing the base field from imaginary multiplicative chaos

TL;DR

This paper demonstrates that the imaginary multiplicative chaos uniquely determines the gradient of the underlying log-correlated Gaussian field, including the 2D Gaussian free field, under mild regularity conditions.

Contribution

It establishes that the imaginary chaos encodes the gradient of the field for a broad class of log-correlated Gaussian fields, extending previous understanding.

Findings

Imaginary chaos determines the gradient of the underlying field.

The result applies to all dimensions d ≥ 2.

The 2D Gaussian free field is measurable with respect to its imaginary 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 $d \geq 2$ 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.