Noise-like analytic properties of imaginary chaos

In this note we continue the study of imaginary multiplicative chaos µβ:= exp(iβΓ), where Γ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of |µβ (Q(x, r))| as r → 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 → 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 ↦→ |µβ (Q(x, r))|2, when normalised additively and multiplicatively, converge as r → 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 µβ (Q(x, r)).