Complex Gaussian Field: Theory & Applications

• Complex Gaussian field is a random field defined by joint complex Gaussian distributions, where the real and imaginary parts exhibit specific statistical independence or symmetry.
• Spectral techniques such as Fourier and Laplace transforms are used to decompose the field and analyze its covariance kernel and moment structure.
• These fields find applications in statistical mechanics, spin glass theory, and high-dimensional probability models, revealing insights into limit theorems and fluctuation phenomena.

A complex Gaussian field is a random field whose finite-dimensional distributions are (jointly) complex Gaussian, typically with structure ensuring that the real and imaginary parts are themselves Gaussian processes, possibly independent or exhibiting particular symmetries. Such fields have become foundational objects in probability, statistical mechanics, mathematical physics, and applied domains such as signal processing and quantum field theory. Their paper encompasses spectral theory, percolation, random geometry, and limit theorems for combinatorial and stochastic models.

1. Fundamental Structure and Definitions

A complex Gaussian field $\{\phi(x)\}_x$ is a collection of complex random variables indexed by a set (often $\mathbb{R}^d$, $\mathbb{Z}^d$, or combinatorial structures like graphs or hypercubes), such that for any finite set of points $(x_1,\ldots,x_n)$, the random vector $(\phi(x_1),\ldots,\phi(x_n))$ is multivariate complex Gaussian. This is characterized by a mean function $m(x)=\mathbb{E}[\phi(x)]$, a covariance kernel $K(x,y)=\mathbb{E}[\phi(x)\overline{\phi(y)}]$, and often a pseudo-covariance $C(x,y)=\mathbb{E}[\phi(x)\phi(y)]$. The field is called “proper” if the pseudo-covariance vanishes, i.e., $\mathbb{E}[\phi(x)\phi(y)] = 0$ for all $x,y$, a property that implies the real and imaginary parts are independent Gaussian fields of equal variance (Boloix-Tortosa et al., 2015). In certain models, the covariance structure is further specified by spectral data, e.g., in terms of the Green function associated with a graph Laplacian or a spatial operator.

2. Spectral, Covariance, and Transform Structures

Complex Gaussian fields often arise as transforms or limits of real Gaussian processes. For instance, in high-dimensional random walks on the hypercube, the field can be realized as a Fourier (or Laplace) transform of a level-set process, leading to a complex field $$\ell^{\mathbf{varkappa}}_\theta = \int_\mathbb{R} e^{i\theta t} \mathbf{varkappa}_t dt$$ whose covariance is $$e^{-\frac{1}{2}(\theta^2+\varphi^2)}\mathbb{E}[e^{\theta\varphi Y}]$$, where $Y$ is a mixing variable induced by an underlying de Finetti-type structure (Griffiths, 20 Oct 2025). Decomposition into real and imaginary components often yields independence: $$\ell^{\mathbf{varkappa}}_\theta = U_\theta + i V_\theta,\quad \text{with} \quad \operatorname{Cov}(U_\theta, V_\varphi) = 0.$$ The covariance kernel, possibly a mixture of bivariate normal forms (but with random correlation arising from structural randomness in the underlying process), is a key analytic object. Such mixtures are characteristic when the generating process is exchangeable or constructed via hierarchical or de Finetti sequences.

More generally, spectral representations (e.g., via Mercer's theorem for integral operators) yield expansions of covariance kernels in terms of eigenvalues and eigenfunctions, with the theoretical structure supporting precise distributional results for high-dimensional moments and limiting behavior (Berschneider et al., 2018).

3. Transformations, Limit Theorems, and Analytic Inversion

Complex Gaussian fields are frequently constructed by analytic transformations—especially Fourier and Laplace transforms—of real Gaussian processes. In the hypercube random walk setting (Griffiths, 20 Oct 2025), the limit field $varkappa_t$ itself arises as the central limit of level-set statistics; the complex field is its Fourier transform. The Plancherel theorem applies, so for suitable integrable processes,

$$varkappa_t = \frac{1}{2\pi} \int_\mathbb{R} e^{-i\theta t} \ell^{\mathbf{varkappa}}_\theta d\theta,$$

and the $L^2$ norm structure is preserved under this isometry.

Decomposition into independent Gaussian variables often exploits spectral theory: the process can be written as a sum over “spin” components (corresponding to the expansion of the random walk or combinatorial model) with independent Gaussian coefficients. For the complex field, this allows explicit computation of moments and spectral analysis via expected exponential functionals.

4. Covariance Structure, Mixtures, and De Finetti Representation

A characteristic feature is the presence of covariance mixing: rather than having a fixed correlation (as in stationary Gaussian fields), the covariance between $\ell^{\mathbf{varkappa}}_\theta$ and $\ell^{\mathbf{varkappa}}_\varphi$ is mixed by a random variable $Y$,

$$\operatorname{Cov}(\ell^{\mathbf{varkappa}}_\theta, \ell^{\mathbf{varkappa}}_\varphi) = e^{-\frac{1}{2}(\theta^2+\varphi^2)} \mathbb{E}[e^{\theta\varphi Y}],$$

where the law of $Y$ encodes the exchangeable or spin glass structure of the original model (with $Y$ being the limit of certain combinations of the local spin variables or given via a point process from the de Finetti measure). Consequently, both the “strength” and “direction” of correlation in the field at different “frequencies” (in the transform picture) are random and determined by an underlying probabilistic structure—fundamentally interlinking probability, combinatorics, and analysis.

Upon decomposition,

$$\operatorname{Cov}(U_\theta, U_\varphi) = e^{-\frac{1}{2}(\theta^2+\varphi^2)} \mathbb{E}[\cosh(\theta\varphi Y)],\quad \operatorname{Cov}(V_\theta, V_\varphi) = e^{-\frac{1}{2}(\theta^2+\varphi^2)} \mathbb{E}[\sinh(\theta\varphi Y)].$$

The independence of $U_\theta$ and $V_\varphi$ underlines the orthogonality in the Hilbert space structure induced by the transform and the symmetry properties of the process.

5. Applications: Spin Glasses, Spectral Analysis, and Statistical Mechanics

The analytic and probabilistic forms of complex Gaussian fields provide simplifications in studying fluctuation phenomena and spectral properties in models of random media, spin glasses, and long-range random walks (Griffiths, 20 Oct 2025). By adopting the transform, off-diagonal and higher-order dependency structures simplify—the covariance function becomes tractable for explicit computations, facilitating the analysis of phase transitions, ergodicity, and fluctuation-dissipation relations.

In connection with mean-field spin glass theory, the field’s decomposition into “k-spin” Hamiltonians and the mixture structure links directly to the classic and generalized Sherrington–Kirkpatrick models, with the transform yielding connections to the overlap structure and ultrametricity.

Through the limit theorems, the field captures universal fluctuation behavior in high-dimensional systems, with the covariance mixing reflecting residual dependencies after the central limit theorem scaling, a phenomenon of foundational importance in probabilistic combinatorics and statistical physics.

6. Connections to Exchangeability, de Finetti Mixtures, and Probabilistic Symmetries

The construction of the complex Gaussian field as a transform of a limit real process strongly reflects the role of exchangeability and de Finetti-type sequences in determining its law. The mixing variable $Y$ embodies the “hidden variable” theory: in the de Finetti representation, the field can be thought of as a mixture over conditional Gaussian laws where the random covariance parameter is determined by the distribution $\nu$ on $[-1,1]$—typically the limit of empirical correlations of the original (e.g., binary) random walk increments.

This perspective interlaces the analytic tractability of Gaussian processes (via their transform methods and explicit kernel formulas) with the deep probabilistic structures (exchangeability, mixtures, spin hierarchies) inherent in random combinatorial and statistical physics systems.

7. Summary Table: Covariance Formulae for the Complex Gaussian Field

Field Component	Covariance Structure	Key Feature
$\ell^{\mathbf{varkappa}}_\theta$	$$e^{-\frac{1}{2}(\theta^2+\varphi^2)} \mathbb{E}[e^{\theta\varphi Y}]$$	Mixture/correlation mixing
$U_\theta$ (real part)	$$e^{-\frac{1}{2}(\theta^2+\varphi^2)} \mathbb{E}[\cosh(\theta\varphi Y)]$$	Even (symmetric) component
$V_\theta$ (imaginary part)	$$e^{-\frac{1}{2}(\theta^2+\varphi^2)} \mathbb{E}[\sinh(\theta\varphi Y)]$$	Odd (antisymmetric) part

These explicit representations underlie both the theoretical analysis and practical computation of process properties, moment estimates, and functional limit theorems.

• Long-range random walks on the hypercube and limit theorems for level-set sums, with detailed discussion of the complex Gaussian field transform, covariance structure, and spectral representations (Griffiths, 20 Oct 2025).
• Eigenvalue decompositions and moment formulas for complex Gaussian quadratic forms, relevant to the L²–norms of complex Gaussian fields (Berschneider et al., 2018).
• Structural analysis of spin glasses and exchangeable sequences in mean-field and combinatorial models, including their Gaussian process representations (Griffiths, 20 Oct 2025).