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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

arXiv:2505.16356v2cond-mat.stat-mechmath.PR

TL;DR

For negative-Hurst fractal Gaussian fields, pointwise definition fails and composite powers are ill-defined, motivating careful study of spatially averaged observables and finite parts. The paper develops an elementary stochastic-process treatment and shows that the quadratic finite part is non-Gaussian and scale-invariant, with Hurst exponent H2 = 2H.

  • Problem

    For H < 0, the field is distribution-valued rather than pointwise, so spatially averaged observables and finite parts of ill-defined composite powers require careful characterization.

    Source: §2 · p. 2

  • Method

    The paper develops an elementary unified stochastic-process treatment in dimension d, focusing on spatial averaging and finite parts of ill-defined composite operators.

    Source: C. Goals and organization of the paper · p. 3

  • Results

    For n = 2, ϕ2 is a non-Gaussian scale-invariant field with H2 = 2H, characterized by arbitrary-order cumulants, a Lévy-Khintchine characteristic function, and anomalous large deviations.

    Source: C. Goals and organization of the paper · p. 3, F. Discussion · p. 18

  • Takeaways & Limitations

    Because these fields are not pointwise functions, their properties are appropriately studied through observables such as spatial averages.

    Source: §2 · p. 2

Abstract

from arXiv · show

The statistical properties of non-linear observables of the fractal Gaussian field $\phi(\vec x)$ 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 $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $. For the special case $n=2$ of quadratic observables, explicit results include the cumulants of arbitrary order, the L\'evy-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 $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$.

I. INTRODUCTION … A. Bra-ket notations to decompose the field |ϕ⟩either in real-space or in Fourier-space

The paper revisits scale-invariant random fields, focusing on negative-Hurst Gaussian fields, spatial-averaging observables, and finite parts of ill-defined composite operators. It develops their statistics using real- and Fourier-space descriptions and stochastic-integral methods.

I. INTRODUCTION

  • The introduction frames scale-invariant random fields as a topic in statistical physics and stochastic processes, and the paper aims to revisit their statistical properties pedagogically.The field’s Hurst exponent distinguishes positive and negative cases with different continuity and stationarity properties.Source: §I · p. 1, A. Scale-invariant random fields of… · p. 1

1. Fractal random fields with Hurst exponent H > 0 : continuous processes with stationary increments

  • Positive-Hurst fields are continuous, with stationary increments rather than translation-invariant field statistics.The variance of their increments grows as a power of distance with exponent 2H > 0.Source: §1 · p. 1

2. Fractal random fields with Hurst exponent H < 0 : stationary fields defined as distributions (and not pointwise)

  • Negative-Hurst fields are translation-invariant but singular at short distances, so they cannot be defined pointwise and must be treated as distributions.Their divergent coincident-point correlations make spatially averaged observables important.Source: §2 · pp. 1–2

1. The important example of the empirical magnetization me associated to the volume Ld in spin models

  • At criticality, long-range correlations invalidate the usual volume-based large-deviation behavior of empirical magnetization, while spatial averaging remains physically relevant.The critical-phenomena example motivates the discussion; the paper thereafter focuses on scale-invariant Gaussian fields.Source: §1 · p. 2

2. Non-linear observables involving scale-invariant Gaussian fields

  • For fractional Gaussian fields, linear observables remain Gaussian, while nonlinear observables—including quadratic ones—are studied using multiple stochastic integrals and Wiener-Ito chaos.The paper presents this material as a self-contained, pedagogical account for statistical physicists.Source: §2 · p. 3, C. Goals and organization of the paper · p. 3

A. Bra-ket notations to decompose the field |ϕ⟩either in real-space or in Fourier-space

C. Goals and organization of the paper

  • Quadratic observables are analyzed through generating functions, cumulants, and the Lévy-Khintchine formula; the finite part ϕ2 is non-Gaussian, with Hurst exponent H2 = 2H and anomalous large deviations.These results include spatial averages of the finite part of the ill-defined operator ϕ2.Source: C. Goals and organization of the paper · p. 3
  • For observables of order n > 2, multiple Ito integrals identify finite parts ϕn of ill-defined composite operators, with scale-invariance exponents Hn = nH and corresponding correlations.The analysis rewrites these observables as order-n observables of white noise.Source: C. Goals and organization of the paper · p. 3

C. Interpretation of the correlation Matrix C ≡E (|ϕ⟩⟨ϕ|) = (−∆)−d 2 −H as a fractional-Laplacian · D. Karhunen–Loeve theorem based on the spectral decomposition of the correlation matrix C

The scale-invariant field’s correlation matrix is identified with a fractional Laplacian, and its square-root spectral decomposition yields a field with white-noise covariance. The Karhunen–Loève construction matches white noise only when the original field is Gaussian; otherwise higher-order correlations distinguish the fields.

C. Interpretation of the correlation Matrix C ≡E (|ϕ⟩⟨ϕ|) = (−∆)−d 2 −H as a fractional-Laplacian

D. Karhunen–Loeve theorem based on the spectral decomposition of the correlation matrix C

  • The Karhunen–Loève construction applies the square root of the correlation matrix to define a transformed field whose correlation matrix coincides with white noise’s.Here the square root is given by the fractional Laplacian identified for the scale-invariant field.Source: D. Karhunen–Loeve theorem based on the… · p. 7
  • If the scale-invariant field is non-Gaussian, the Karhunen–Loève transformed field is also non-Gaussian and differs from white noise through higher-order correlations.Source: D. Karhunen–Loeve theorem based on the… · p. 7
  • If the scale-invariant field is Gaussian, its Karhunen–Loève transformed field coincides with white noise, allowing its statistics to be reformulated in white-noise terms.Source: D. Karhunen–Loeve theorem based on the… · p. 7

E. Consequences for linear and quadratic observables related to spatial-averaging over a volume Rd … 4. Empirical spatial-average of the fluctuating part ϕ2(⃗x) = ϕ2(⃗x) −E(ϕ2(⃗x)) of the ill-defined composite operator ϕ2(⃗x)

The section develops spatial-averaging observables from a kernel defined by a shape and scale, then considers the empirical magnetization and the spatial average of the fluctuating part of the quadratic field. Spatial averaging preserves the field’s Hurst exponent for magnetization, while the quadratic observable addresses the divergence of the uncentered composite operator.

1. General properties of linear and quadratic observables

  • Linear observables have zero averaged value, and their variances can be computed from the correlation matrix C in real or Fourier space.Source: §1 · p. 7
  • Spatial averaging with a kernel based on a centered shape A(X) constructs observables over volumes scaling as R^d.The kernel is introduced for use in constructing spatial-averaged observables.Source: §1 · pp. 8–9
  • The Gaussian averaging shape is smooth, rotation-invariant, and retains a Gaussian Fourier transform; its kernel corresponds to a heat kernel with 4πt = R^2.Source: §1 · p. 9

3. Empirical magnetization MR associated to the volume Rd around the origin

  • The empirical magnetization is statistically scale-invariant with the field’s Hurst exponent H, and its variance depends on the shape and field correlation.Source: §3 · p. 9
  • Spatial averaging leaves H stable, and the short-distance empirical magnetization Mϵ at R = ϵ can regularize the field.The passage also presents regularization as a way to understand calculations involving the field.Source: §3 · p. 9

4. Empirical spatial-average of the fluctuating part ϕ2(⃗x) = ϕ2(⃗x) −E(ϕ2(⃗x)) of the ill-defined composite operator ϕ2(⃗x)

  • Because the divergence makes E(ϕ^2) infinite, the section introduces the fluctuating part ϕ^2 − E(ϕ^2) as a new field.Source: §4 · p. 10
  • Choosing a real-space diagonal quadratic observable with the spatial-averaging kernel yields the spatial average of the fluctuating quadratic field over a volume scaling as R^d.Source: §4 · p. 10

F. Discussion … C. Gaussian statistics of linear observables with the example of the empirical magnetization MR

For Gaussian fractional Gaussian fields, the correlation matrix determines the full statistics, and linear observables inherit Gaussian statistics. For empirical magnetization, large deviations scale with the Hurst exponent rather than the usual volume exponent, while white noise provides a construction of the fields.

F. Discussion

  • The discussion turns to fractional Gaussian fields to obtain explicit results on questions involving higher correlations.The preceding discussion considered properties depending only on the correlation matrix.Source: F. Discussion · p. 10

A. Gaussian probability distribution based on the inverse C−1 = (−∆)

  • The correlation function’s simple power-law validity range is complementary to that of the Gaussian kernel, as expected for inverse matrices.The kernel is the inverse of the correlation matrix.Source: A. Gaussian probability distribution… · p. 11

B. Link with the White-Noise W(.) in dimension d with the negative Hurst exponents H = −d

  • Fractional Gaussian fields can be constructed from white noise by applying the appropriate fractional Laplacian, which rescales Fourier components.The white-noise representation also permits reconstruction of the field from real-space components by convolution.Source: B. Link with the White-Noise W(.) in… · p. 11

C. Gaussian statistics of linear observables with the example of the empirical magnetization MR

  • A Gaussian field’s linear observable is Gaussian, with its distribution and generating function determined by its variance.The empirical magnetization probability distribution is considered when averaging over a volume scaling as R^d.Source: C. Gaussian statistics of linear… · p. 12
  • The empirical magnetization’s large-deviation exponential scales as R^-2H rather than R^d, recovering the usual volume scaling only at H = -d/2.For the free Gaussian field example, the corresponding unusual exponent is R^(d-2) rather than R^d.Source: C. Gaussian statistics of linear… · p. 12
  • For d > 2, the free Gaussian field ϕf has correlation matrix corresponding to the inverse (−∆)−1 of the opposite Laplacian; its real-space matrix elements are given by a power law.In d = 1, the free Gaussian field corresponds to Brownian motion B(x) with positive Hurst exponent H = 1/2.Source: C. Gaussian statistics of linear… · p. 12

IV. STATISTICS OF QUADRATIC OBSERVABLES B = ⟨ϕ|B|ϕ⟩OF THE FRACTIONAL-GAUSSIAN-FIELD · A. Computation of the generating function E · C. Infinite-divisibility and the L´evy-Khintchine formula for the characteristic function E

The section studies quadratic observables of a fractional Gaussian field using a generating function, cumulants, and the characteristic function. It expresses cumulants through traces involving the correlation and observable operators, and gives a Lévy–Khintchine representation under a discrete positive-eigenvalue assumption.

IV. STATISTICS OF QUADRATIC OBSERVABLES B = ⟨ϕ|B|ϕ⟩OF THE FRACTIONAL-GAUSSIAN-FIELD

  • The quadratic observables considered are parametrized by a symmetric operator B.Source: §IV · p. 13

A. Computation of the generating function E

  • Changing variables to white noise expresses the generating function through a trace of log(1 − 2λF), which is used to compute cumulants and analyze infinite divisibility.Source: A. Computation of the generating… · p. 13
  • For every order n, the cumulant of B involves tr((BC)^n), with C the field correlation operator and B the observable operator.The trace-power expression follows from the generating-function expansion and cyclicity of the trace.Source: A. Computation of the generating… · p. 13
  • For the order-n trace evaluation, real space uses an integral over 2n variables, whereas Fourier space uses n variables.Source: A. Computation of the generating… · p. 14

C. Infinite-divisibility and the L´evy-Khintchine formula for the characteristic function E

D. Rephrasing for the fluctuations of the operator |ϕ⟩⟨ϕ| around its averaged-value E (|ϕ⟩⟨ϕ|) = C · E. Application to the spatial-average of the fluctuating part ϕ2(⃗x) of the composite operator ϕ2(⃗x) · 1. Scaling properties of the cumulants

The section rewrites fluctuations around the correlation matrix and applies that framework to spatial averages of the fluctuating part of ϕ². Their cumulants scale with R, while their large deviations retain an unusual exponent except in the white-noise case.

D. Rephrasing for the fluctuations of the operator |ϕ⟩⟨ϕ| around its averaged-value E (|ϕ⟩⟨ϕ|) = C

  • Fluctuations of B are expressed through |ϕ⟩⟨ϕ|−C, with the generating function expanded in cumulants of order n≥2 and its characteristic function written in Lévy–Khintchine form.The Lévy–Khintchine representation uses the density νF(u).Source: D. Rephrasing for the fluctuations of… · pp. 15–16

E. Application to the spatial-average of the fluctuating part ϕ2(⃗x) of the composite operator ϕ2(⃗x)

1. Scaling properties of the cumulants

  • The cumulants of arbitrary order n≥2 scale as R^(n2H), and the real-space correlation of ϕ² is the square of the initial field’s correlation, making ϕ² a non-Gaussian scale-invariant field.The cumulant prefactor involves an integral over n variables.Source: §1 · pp. 16–17
  • The integral in Eq. 110 diverges in the lower-half range −d/2 < H < −d/4.Source: §1 · p. 17
  • Rescaled eigenvalues Fα recover the R^(n2H) cumulant scaling, while the characteristic function depends only on Θ=θR^(2H).Source: §1 · p. 18
  • For large R, large deviations have an exponential exponent scaling as R^(−2H); usual volume scaling R^d is recovered only at H=−d/2, the white-noise case.Source: §1 · p. 18

F. Discussion … D. Interpretation of the Ito integral In(F (n)) in terms of the initial field ϕ

The discussion characterizes the quadratic observable as a scale-invariant non-Gaussian field, then develops a Wiener–Ito framework for higher-order observables and interprets its Ito integrals in terms of the initial field.

F. Discussion

  • The quadratic observable ϕ2 is a scale-invariant non-Gaussian field with Hurst exponent H2 = 2H, characterized by arbitrary-order cumulants or its Levy-Khintchine characteristic function.The discussion turns next to observables of order n > 2.Source: F. Discussion · p. 18

V. STATISTICS OF OBSERVABLES OF ORDER n > 2 OF THE FRACTIONAL-GAUSSIAN-FIELD

A. Observable B(n) = ⟨B(n)|ϕ⊗n⟩of order n parametrized by the symmetric tensor B(n)(⃗x1, ..,⃗xn)

D. Interpretation of the Ito integral In(F (n)) in terms of the initial field ϕ

1. Example of the cubic Ito integral I3(F (3)) … VI. CONCLUSIONS

The paper uses Wiener–Ito integrals to construct finite parts of ill-defined composite operators and study their spatial averages and correlations. It also summarizes explicit quadratic-observable results, including arbitrary-order cumulants, the characteristic function, and anomalous large deviations.

1. Example of the cubic Ito integral I3(F (3))

  • For cubic observables, the Ito-integral construction uses a symmetrized product with pairwise covariance terms subtracted.The passage identifies this construction for n = 3.Source: §1 · p. 21

2. Example of the quartic Ito integral I4(F (4))

  • For quartic observables, the Ito-integral example likewise uses a symmetrized product with covariance contractions subtracted and added.The displayed expression includes subtractions involving one covariance and additions involving products of two covariances.Source: §2 · p. 21

E. Application to the spatial-average of the finite part ϕn(⃗x) of the composite operator ϕn(⃗x)

  • The naive spatial average of the ill-defined composite operator has divergent constituent functions because the field correlation diverges at coinciding points.The passage states that the functions diverge for any l ≥ 1 as y approaches x.Source: E. Application to the spatial-average… · pp. 21–22
  • The Ito integral gives a finite linear combination of observables that would each diverge individually, representing the spatial average of the composite operator’s finite part.The construction is expressed using the spatial-averaging kernel A_R.Source: E. Application to the spatial-average… · p. 22
  • The finite-part correlation involves the nth power of the original field correlation C(x1,x2), generalizing the n = 2 case.The variance of the Ito integral is given as the basis for this correlation result.Source: E. Application to the spatial-average… · pp. 22–23
  • The finite part is a non-Gaussian scale-invariant field, and the convergence region for the original correlation shrinks as n grows.The supplied passage also gives the condition 2n < H < 0, which is not restated here.Source: E. Application to the spatial-average… · p. 23

F. Generalization to arbitrary observables of the Fractional-Gaussian-Field field ϕ

  • Functionals of the white noise can be expanded into multiple Ito integrals through the Wiener–Ito chaos expansion, extending the treatment to observables of arbitrary order.The expansion is reinterpreted in terms of the fractional Gaussian field.Source: F. Generalization to arbitrary… · p. 23

VI. CONCLUSIONS

  • For quadratic observables, the paper gives arbitrary-order cumulants, a Lévy–Khintchine characteristic function, and anomalous large-deviation properties; for n > 2, it uses multiple Ito integrals to identify finite parts and compute correlations involving H_n = nH.The conclusions describe the analysis for negative Hurst exponent H < 0 in dimension d.Source: §VI · p. 23

Appendix A: fractional Gaussian fields with positive or negative Hurst exponents in dimension d = 1 … 1. Multiple stochastic integrals of order n involving the white noise W(.)

The appendices connect one-dimensional Brownian and fractional Brownian processes to fractional Gaussian noise with negative Hurst exponent, then introduce multiple stochastic Ito integrals as a basis for analyzing white-noise functionals. They distinguish stationary increments from stationary noise and outline the higher-dimensional spatial-averaging connection.

Appendix A: fractional Gaussian fields with positive or negative Hurst exponents in dimension d = 1

  • Appendix A recalls Brownian motion, fractional Brownian motion, and their associated Gaussian noises to connect positive and negative Hurst exponents in one dimension.It also links the one-dimensional discussion to the arbitrary-dimensional case treated in the main text.Source: Appendix A · p. 23

1. The Brownian motion B(x) of positive Hurst exponent H = 1 2 in dimension d = 1

  • Brownian motion is defined by integrating one-dimensional white noise from a chosen origin, while its increments—not the process itself—are translation invariant.The usual choice is B(0) = 0; the white noise is translation invariant.Source: §1 · pp. 23–24

2. The fractional Brownian motion BH(x) of positive Hurst exponent 0 < H < 1 in dimension d = 1

  • Fractional Brownian motion has translation-invariant increments, motivating the associated fractional Gaussian noise defined by differentiation.The noise correlation also depends only on separation, consistent with its translation invariance.Source: §2 · pp. 24–25

b. Region 1

  • For 1/2 < H < 1, fractional Gaussian noise has a power-law correlation with amplitude H(2H − 1), expressible using H′ = H − 1.The passage relates this negative-Hurst-exponent form to power-law decaying correlations in arbitrary dimension.Source: b. Region 1 · p. 25

5. Generalizations to higher dimension d > 1

  • In higher dimensions, a fractional Brownian field generalizes the one-dimensional process, and its spatial average corresponds to the empirical magnetization for a negative-Hurst field.The one-dimensional ratio’s variance involves H′ = H − 1; in the generalization, the spatial-averaging kernel is AR(x).Source: §5 · p. 26

Appendix B: Reminder on the Wiener-Ito orthogonal basis for functionals of the White Noise

  • Appendix B gives a self-contained introduction to the Wiener-Ito chaos expansion because its multiple stochastic integrals have a long mathematical history but are less familiar in physics..Source: Appendix B · p. 26

1. Multiple stochastic integrals of order n involving the white noise W(.)

  • Ito integrals have zero mean for every order n ≥ 1, whereas symmetric integrals of even order can have nonzero means through pairings of their arguments.For order 2l, the Wick theorem pairs the 2l positions, yielding the nonvanishing mean described in the passage.Source: §1 · p. 27

a. Stochastic integrals of order n = 1 … 3. Rephrasing as the orthogonality of the family |W [Ito]

Multiple Ito integrals are orthogonal across different orders, with same-order correlations determined by symmetric-kernel inner products; this structure supports Wiener-Ito expansions of white-noise functionals and stochastic integrals.

a. Stochastic integrals of order n = 1

  • For n = 1, the correlation between Ito integrals is the scalar product of their associated functions.The section also notes that the variance reduces to the corresponding n = 1 expression.Source: a. Stochastic integrals of order n = 1 · p. 27

b. Stochastic integrals of order n = 2

  • For n = 2, I2(f2) has vanishing average, whereas S2(f2) has a non-vanishing average that reduces to the trace of f2.The passage also gives the variance of I2(f2), without specifying its formula in the supplied text.Source: b. Stochastic integrals of order n = 2 · p. 27

a. Correlation between two Ito integrals In(fn) and In(gn) of arbitrary order n

  • For symmetric kernels of arbitrary order n, the correlation of In(fn) and In(gn) is n! times their inner product.The variance of In(fn) follows as a special case of this same-order correlation result.Source: a. Correlation between two Ito… · p. 28

b. Vanishing correlations between Ito integrals In(.) and Im(.) of different orders n̸ = m

c. Conclusion on the orthogonality properties of the multiple Ito integrals In(.)

  • Adding the order-zero constant lets the properties be summarized together: positive-order Ito integrals have vanishing averages, different orders have zero correlation, and same-order correlations follow the inner-product rule.Source: c. Conclusion on the orthogonality… · p. 28

3. Rephrasing as the orthogonality of the family |W [Ito]

  • Any white-noise functional can be expanded on the complete orthogonal family of Ito functionals, with expansion coefficients computed from correlations with that family.The order-zero coefficient is the functional’s average.Source: §3 · p. 29
  • The Wiener-Ito expansion of Sn(fn) consists of Ito integrals of orders n−2l associated with partial traces of fn, with the order-n Ito integral as its first term.For even n, the final term is a constant and accounts for the average; the other Ito integrals have vanishing averages.Source: §3 · p. 31

d. Examples for small values of n = 2, 3, 4

The examples for n = 2, 3, and 4 show how spatial-averaging observables decompose into Wiener-Ito integrals and how their variances relate to the corresponding integral formulas and Wick pairings.

d. Examples for small values of n = 2, 3, 4

6. Wiener-Ito chaos-expansion of the exponential functional eλI1(h) = eλ⟨h|W ⟩= eλ · a. Usual expansion of the exponential functional eλI1(h) in terms of stochastic integrals Sn(.) · b. Wiener-Ito Chaos-expansion of the exponential functional eλI1(h) in terms of Ito integrals In(.)

The section compares the usual stochastic-integral expansion of the exponential white-noise functional with its Wiener-Ito chaos expansion. The functional’s averaged value is the generating function of I1(h), with a simple explicit expression.

6. Wiener-Ito chaos-expansion of the exponential functional eλI1(h) = eλ⟨h|W ⟩= eλ

  • The averaged exponential functional gives the generating function of I1(h), for which the passage notes a simple explicit expression.Source: §6 · p. 33

a. Usual expansion of the exponential functional eλI1(h) in terms of stochastic integrals Sn(.)

b. Wiener-Ito Chaos-expansion of the exponential functional eλI1(h) in terms of Ito integrals In(.)

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