Imaginary Multiplicative Chaos

• Imaginary multiplicative chaos is defined as the renormalized exponential of a log‐correlated Gaussian field with an imaginary coefficient, leading to a random generalized function rather than a measure.
• Its rigorous construction via mollification and Wick renormalization provides precise regularity characterization in Sobolev and Besov spaces, with deterministic scaling laws.
• The theory connects probability, conformal field theory, and statistical physics, underpinning scaling limits in models such as the XOR–Ising and sine–Gordon models.

Imaginary multiplicative chaos is a central object in modern probability, mathematical physics, and complex analysis, arising as the renormalized exponential of a log-correlated Gaussian field with an imaginary coefficient. Unlike its real counterpart, which produces random (multifractal) measures, imaginary multiplicative chaos is fundamentally a random generalized function, exhibiting oscillatory rather than positive fluctuations. This object encodes fine analytic and geometric properties of planar models, plays a key role in the scaling limits of statistical physics (notably in the XOR–Ising and sine–Gordon models), connects to conformal field theory and Liouville quantum gravity, and admits rigorous construction and regularity characterization via functional analysis, stochastic analysis, and operator theory.

1. Construction and Formalism

Let $X$ be a real-valued log-correlated Gaussian field on a domain $U \subset \mathbb{R}^d$, with covariance structure

$$\mathbb{E}[X(x)X(y)] = \log \frac{1}{|x-y|} + g(x,y)$$

where $g$ is bounded and continuous locally. The "imaginary multiplicative chaos" is formally defined as

$\mu_\beta(x) = :\!\exp(i\beta X(x))\!:$

where $:\!\cdot\!:$ denotes the Wick renormalization, and $\beta \in \mathbb{R}$ is a parameter (the "chaos intensity"). Because $X$ is only a generalized function (not pointwise defined), the exponential is not directly meaningful. The standard construction proceeds via approximation:

• Regularize $X$ by convolution: $X_\varepsilon(x)$ for mollifier scale $\varepsilon > 0$.
• Define the approximating fields:

$$\mu_{\beta,\varepsilon}(x) = \exp\big(i\beta X_\varepsilon(x) + \frac{\beta^2}{2} \mathbb{E}[X_\varepsilon(x)^2]\big)$$

• The renormalized chaos $\mu_\beta$ is then obtained as the $\varepsilon\to 0$ limit in an appropriate negative Sobolev or Besov space.

In the limit, $\mu_\beta$ becomes a stationary random distribution. No infinite divisibility or measure-valued structure survives, and all constructions are only meaningful as generalized functions or distributions, not as random measures (Junnila et al., 2018, Junnila et al., 2019).

2. Regularity and Functional-Analytic Properties

The precise functional space where imaginary multiplicative chaos lives is determined by the covariance and the parameter $\beta$. The key quantification is in terms of Besov and Sobolev space regularity:

• If $X$ is as above and $0 < \beta < \sqrt{d}$, then for any $s < -d/2$, $\mu_\beta \in H^s_{\text{loc}}(\mathbb{R}^d)$ almost surely.
• Finer regularity is characterized in Besov spaces: almost surely, $\mu_\beta \in B^s_{p,q}$ for any $s < -\beta^2/2$ and $1 \leq p, q \leq \infty$, but not for $s > -\beta^2/2$ (Junnila et al., 2018, Junnila et al., 2019, Aru et al., 26 Jan 2024). At the critical exponent $s = -\beta^2/2$, inclusion holds if and only if $p < \infty$, $q = \infty$.

Thus, the optimal regularity is strictly less than what would be allowed for white noise, reflecting the high oscillatory roughness induced by the logarithmic singularity and the imaginary exponent.

3. Stochastic Properties: Moments, Tails, and Density

When integrated against test functions $f \in C^\infty_c(U)$, $\mu_\beta(f) = \int f(x) \mu_\beta(x)\,dx$ defines a complex-valued random variable with the following properties:

• All moments exist and are finite for any order $p > -2$ and $0 < \beta < \sqrt{d}$ (Junnila et al., 2018, Junnila et al., 2019, Aru et al., 8 Mar 2024).
• The law of $\mu_\beta(f)$ admits a Schwartz-class (smooth, rapidly decaying) density with respect to Lebesgue measure on $\mathbb{C}$. This density is strictly positive everywhere (Aru et al., 2020, Aru et al., 8 Mar 2024).
• Tail estimates and moment growth are sub-exponential in moment order. For example, the $2N$th moment has growth bounds

$$\mathbb{E}|\mu_\beta(f)|^{2N} \leq \|f\|_\infty^{2N} C^N N^{\beta^2 N/d}$$

for some $C>0$.

Applications of Malliavin calculus yield both existence and strict positivity of the density (Aru et al., 2020, Aru et al., 8 Mar 2024). Operator-theoretic methods are essential for establishing uniform small ball probabilities and Sobolev norm estimates for negative moments.

4. Local Laws, Monofractality, and Scaling Behavior

The fine-scale structure of imaginary multiplicative chaos is characterized by precise monofractal scaling:

• For any $x$ and small $r>0$, consider the "local mass" $\mu_\beta(Q(x,r))$ where $Q(x,r)$ is a square of sidelength $2r$ centered at $x$.
• Almost surely,

$$\lim_{r \to 0} \frac{\log |\mu_\beta(Q(x,r))|}{\log r} = 2 - \frac{\beta^2}{2}$$

i.e., the scaling exponent is deterministic and site-independent: monofractality (Aru et al., 26 Jan 2024).

Furthermore, a law of the iterated logarithm holds: $$\limsup_{r\to 0} \frac{|\mu_\beta(Q(x,r))|}{r^{2 - \beta^2/2} (\log\log(1/r))^{\beta^2/4}} = c^*(\beta)^{- \beta^2 / 4}$$ for some $c^*(\beta)>0$.

The collection of exceptional (fast) points, with abnormally large limsup, has a Hausdorff dimension $2- c^*(\beta) a^{4/\beta^2}$. These multifractal properties parallel those known for thick points of the Gaussian free field, but differ in details due to the oscillatory (not positive) structure.

5. Connections to Statistical Physics and Conformal Field Theory

Imaginary multiplicative chaos appears naturally in the scaling limits of critical statistical mechanics models:

• The real part, $\cos(\beta X(x))$, arises in the scaling limit of the spin field for the critical planar XOR–Ising model (Junnila et al., 2018). The scaling limit is

$$\delta^{-1/4} \int_U f(x) \mathcal{S}_\delta(x)\,dx \to \mathcal{C}^2 \int_U f(x) (\frac{2|\varphi'(x)|}{\Im\varphi(x)})^{1/4} \cos(\frac{1}{\sqrt{2}} X(x))\,dx$$

where $\mathcal{C}$ is explicit, $X$ is the Gaussian free field, and $\varphi$ is a conformal map.

• With magnetic perturbation, the limit involves $\cos(\frac{1}{\sqrt{2}} X_{\text{sG}}(x))$ where $X_{\text{sG}}$ is the sine–Gordon field.

In conformal field theory and Liouville quantum gravity, complex Gaussian multiplicative chaos is employed to construct rigorously vertex operators such as the "tachyon fields" $e^{\gamma X + i\beta Y}$, whose scaling dimensions satisfy the KPZ formula (Lacoin et al., 2013).

In these field theory contexts, the imaginary chaos serves as the mathematical realization of highly oscillatory, non-measure-valued fields, and their real parts correspond to scaling limits of lattice models.

6. White-Noise Limits and Angular Structure

A distinctive phenomenon is that, after normalization, the modulus $|\mu_\beta(Q(x,r))|^2$ converges in law, but not in probability, to white noise as $r \to 0$ (Aru et al., 26 Jan 2024):

• Define

$$W_r(x) := r^{\beta^2 - 5}\big(|\mu_\beta(Q(x, r))|^2 - \mathbb{E}|\mu_\beta(Q(x, r))|^2\big)$$

Then $W_r$ converges in law in $H^{-2}_0([0,1]^2)$ to a random distribution whose second derivative is spatial white noise.

Consequently, the macroscopic field built from the squared modulus is universal and noise-like; all non-trivial dependence and structure is encoded in the angular part (the argument) of $\mu_\beta$, not its magnitude.

7. Reconstruction and Informational Content

It is possible to reconstruct the gradient (and, up to additive constant, the entire field) $X$ from its imaginary chaos $:\!\exp(i\beta X)\!:$ (Aru et al., 2020):

• For log-correlated $X$ with covariance $-\log|x-y| + g(x,y)$, and regular $g$, $\exp(i\beta X)$ determines $\nabla X$ in the sense of distributions.
• In particular, for the zero-boundary Gaussian free field in the plane, $X$ is measurable with respect to its imaginary multiplicative chaos.

This distinguishes imaginary chaos from real chaos, which loses information about additive constants due to positivity and measure-valued structure.

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Imaginary multiplicative chaos thus provides the rigorous mathematical underpinning for highly oscillatory, scale-invariant random distributions, while exhibiting rich analytic, geometric, and probabilistic behaviors. It mediates between probability, conformal field theory, and the fine structure of random fields, providing connections between scaling limits, fractal geometry, and the algebra of field-theoretic operators via precise chaos expansions, regularity theory, and moment analysis (Junnila et al., 2018, Junnila et al., 2019, Aru et al., 26 Jan 2024, Aru et al., 2020, Aru et al., 8 Mar 2024).