Noise-like analytic properties of imaginary chaos

TL;DR

The paper analyzes imaginary multiplicative chaos $\mu_\beta = e^{i\beta\Gamma}$ for the 2D Gaussian free field, focusing on the fine-scale analytic properties of $|\mu_\beta(Q(x,r))|$ as $r\to0$. It presents a coherent set of results: (i) monofractality and a sharp Hölder regularity bound for $\mu(Q(x,r))$, (ii) a law of the iterated logarithm with a universal constant $c^*(\beta)$ and a fractal description of fast points, (iii) the exact Besov regularity at the critical index $s=-\beta^2/2$, (iv) a limiting white-noise regime for the centered, renormalized squared mass $|\mu(Q(x,r))|^2$, and (v) a precise analysis of the limiting field from a probabilistic and harmonic-extension perspective. The results illuminate the noise-like character of imaginary chaos, show information loss in the modulus in the limit, and reveal that angular information encodes the underlying field. Methodologically, the work combines Malliavin-based density arguments, Sobolev/Besov techniques, wavelet analysis, geometric decompositions, and delicate probabilistic controls (Onsager-type inequalities, harmonic extensions, and LIL arguments) to obtain a comprehensive picture. Overall, the paper advances the understanding of the fine-scale structure of imaginary chaos and its connections to white noise limits and fractal geometry.

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 $μ_β(Q(x,r))$.

Theorems & Definitions (54)

• Proposition 1.1: Hölder regularity
• Theorem 1.2: Monofractality
• Remark 1.3
• Theorem 1.4: Law of the iterated logarithm
• Theorem 1.5: Exceptional points
• Theorem 1.6: Leble2017
• Theorem 1.7: Regularity of imaginary chaos
• Theorem 1.8
• Lemma 2.1
• proof