Statistical properties of non-linear observables of fractal Gaussian fields with a focus on spatial-averaging observables and on composite operators

The statistical properties of non-linear observables of the fractal Gaussian field $ϕ(\vec x)$ of negative Hurst exponent $H<0$ in dimension $d$ are revisited with a focus on spatial-averaging observables and on the properties of the finite parts $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec x) $. For the special case $n=2$ of quadratic observables, explicit results include the cumulants of arbitrary order, the Lévy-Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order $n>2$ is analyzed via the Wiener-Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts $ϕ_n(\vec x)$ of the ill-defined composite operators $ϕ^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$.

💡 Research Summary

This paper provides a comprehensive and pedagogical analysis of the statistical properties of non-linear observables derived from fractal Gaussian fields with a negative Hurst exponent (H < 0) in d dimensions. The central challenge addressed is that such fields, while being statistically scale-invariant and stationary, are too singular to be defined pointwise and must be interpreted as tempered distributions. This renders composite operators like φ^n(x) ill-defined. The paper’s primary objectives are to study spatially-averaged observables and to identify and characterize the “finite parts” φ_n(x) of these divergent composite operators.

The introduction contrasts fields with H > 0 (e.g., fractional Brownian motion, continuous with stationary increments) and H < 0 (e.g., critical spin fields, stationary but distribution-valued). For the latter, spatial averaging over a region (like the empirical magnetization in spin models) becomes a physically essential observable, exhibiting anomalous large deviation properties at criticality due to long-range correlations.

Section II sets up the formalism using bra-ket notation from quantum mechanics to describe the field in real and Fourier space. The scale-invariance is encoded in the correlation matrix C, whose elements decay as a power law C(x, y) ~ |x-y|^{-2H}. This matrix is shown to be interpretable as a fractional Laplacian operator, C = (-Δ)^{-d/2 - H}.

In Section III, the focus is on the specific case where φ is a Gaussian field. Here, linear observables (like the spatial average over a volume) remain Gaussian. However, due to the long-range power-law correlations, their large deviation behavior deviates from the standard form ~exp