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

Cecile Monthus

TL;DR

This work analyzes non-linear observables of fractal Gaussian fields with negative Hurst exponent H in dimension d, focusing on spatial-averaging observables and the finite parts φ_n of ill-defined composite operators φ^n. It builds a coherent framework around the correlation matrix C = E(|φ><φ|), showing C = (−Δ)^{−(d/2+H)} and employing Karhunen–Loève decomposition to connect the field to white noise; for Gaussian φ, this reduces to a White-Noise-based description that yields exact results for linear and quadratic observables, including anomalous large deviations for the empirical magnetization and the spatial average of φ^2. The paper develops a Wiener–Ito chaos expansion for arbitrary order n to construct finite parts φ_n via Ito integrals, revealing that the finite parts obey H_n = nH and possess power-law correlations with exponent −2nH, thereby highlighting non-Gaussianity at all n>1. The results provide a practical, transparent methodology for computing cumulants, characteristic functions, and large-deviation tails of spatially averaged, non-linear functionals, with implications for critical phenomena and scale-invariant Gaussian models where negative H arise. These contributions unify real-space and Fourier-space analyses and offer a concrete toolbox for evaluating renormalized composite operators in fractal Gaussian fields.

Abstract

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 $H_n=nH$.

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

TL;DR

This work analyzes non-linear observables of fractal Gaussian fields with negative Hurst exponent H in dimension d, focusing on spatial-averaging observables and the finite parts φ_n of ill-defined composite operators φ^n. It builds a coherent framework around the correlation matrix C = E(|φ><φ|), showing C = (−Δ)^{−(d/2+H)} and employing Karhunen–Loève decomposition to connect the field to white noise; for Gaussian φ, this reduces to a White-Noise-based description that yields exact results for linear and quadratic observables, including anomalous large deviations for the empirical magnetization and the spatial average of φ^2. The paper develops a Wiener–Ito chaos expansion for arbitrary order n to construct finite parts φ_n via Ito integrals, revealing that the finite parts obey H_n = nH and possess power-law correlations with exponent −2nH, thereby highlighting non-Gaussianity at all n>1. The results provide a practical, transparent methodology for computing cumulants, characteristic functions, and large-deviation tails of spatially averaged, non-linear functionals, with implications for critical phenomena and scale-invariant Gaussian models where negative H arise. These contributions unify real-space and Fourier-space analyses and offer a concrete toolbox for evaluating renormalized composite operators in fractal Gaussian fields.

Abstract

The statistical properties of non-linear observables of the fractal Gaussian field of negative Hurst exponent in dimension are revisited with a focus on spatial-averaging observables and on the properties of the finite parts of the ill-defined composite operators . For the special case 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 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 of the ill-defined composite operators and to compute their correlations involving the Hurst exponents .

Paper Structure

This paper contains 72 sections, 231 equations.

Table of Contents

  1. Introduction
  2. Scale-invariant random fields of Hurst exponent $H$ either positive $H>0$ or negative $H<0$
  3. Fractal random fields with Hurst exponent $H>0$ : continuous processes with stationary increments
  4. Fractal random fields with Hurst exponent $H<0$ : stationary fields defined as distributions (and not pointwise)
  5. Spatial-averaged observables involving scale-invariant fields of Hurst exponent $H<0$
  6. The important example of the empirical magnetization $m_e$ associated to the volume $L^d$ in spin models
  7. Non-linear observables involving scale-invariant Gaussian fields
  8. Goals and organization of the paper
  9. Correlation matrix ${\bold C}$ for scale-invariant fields with Hurst exponents $H<0$
  10. Bra-ket notations to decompose the field $\vert \phi \rangle$ either in real-space or in Fourier-space
  11. Definition of a scale-invariant field $\phi$ with negative Hurst exponent $H \in ]-\frac{d}{2} ,0[$ in dimension d
  12. Interpretation of the correlation Matrix ${\bold C} \equiv {\mathbb E} \left( \vert \phi \rangle \langle \phi \vert \right) = (-{\bold \Delta})^{- \frac{d}{2}-H}$ as a fractional-Laplacian
  13. Reminder on the Laplacian operator ${\bold \Delta}$ and on its fractional-powers $(-{\bold \Delta})^{- \frac{\alpha}{2}}$
  14. Correspondance with the correlation matrix ${\bold C}$ of the scale-invariant field via $\alpha=d+2H$
  15. Karhunen–Loeve theorem based on the spectral decomposition of the correlation matrix ${\bold C}$
  16. ...and 57 more sections