Noise-like analytic properties of imaginary chaos

Abstract

In this note we continue the study of imaginary multiplicative chaos $μ_β:= \exp(i βΓ)$, where $Γ$ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of $|μ_β(Q(x,r))|$ as $r \to 0$, where $Q(x,r)$ is a square of side-length $2r$ centred at $x$. More precisely, we prove monofractality of this process, a law of the iterated logarithm as $r \to 0$ and analyse its exceptional points, which have a close connection to fast points of Brownian motion. Some of the technical ideas developed to address these questions also help us pin down the exact Besov regularity of imaginary chaos, a question left open in [JSW20]. All the mentioned properties illustrate the noise-like behaviour of the imaginary chaos. We conclude by proving that the processes $x \mapsto |μ_β(Q(x,r))|^2$, when normalised additively and multiplicatively, converge as $r \to 0$ in law, but not in probability, to white noise; this suggests that all the information of the multiplicative chaos is contained in the angular parts of $μ_β(Q(x,r))$.