Scale-Dependent Field Transformation

• Scale-Dependent Field Transformation is a phenomenon where different rescaling rates yield varying statistical properties and anisotropic behaviors.
• The theory distinguishes between subcritical, critical, and supercritical regimes, resulting in limits like fractional Brownian sheets and operator-scaling Gaussian fields.
• Analytical methods such as homogenization techniques and multifractal analysis provide practical insights for modeling multiscale random fields and physical systems.

A scale-dependent field transformation is a structural change in the limiting behavior, dependence, or statistical properties of stochastic or deterministic fields under rescaling operations that act differently across distinct spatial, temporal, or parameter axes. Such transformations typically emerge when systems exhibit anisotropy or multifractality, and are mathematically identified via distinct scaling limits, transitions between universality classes, or the emergence of new types of operator-scaling fields. This article synthesizes recent developments in this area, grounded in the theory of partial sums of random fields, multifractal stochastic processes, and anisotropic homogenization.

1. Anisotropic Scaling and Spectral Singularity Frameworks

Scale-dependent transformations were formalized for stationary random fields on lattices such as $\mathbb{Z}^2$ with anisotropic long-range dependence (LRD), negative dependence (ND), or mixed long-range negative dependence (LRND). Such fields exhibit a power-law singularity in their spectral density near the origin of the form

$$f(x) \approx \begin{cases} \rho(x)^{-1} L(x), & \text{LRD}, \ \rho(x) L(x), & \text{ND}, \ \frac{L(x)}{|x_1|^{\mu_1} \rho(x)^{1+(\mu_2/\alpha_2)}}, & \text{LRND}, \ |x_1|^{-\alpha_1} |x_2|^{-\alpha_2} L(x), & \text{hyperbolic}, \end{cases}$$

with the anisotropic radial function $\rho(x) = |x_1|^{\alpha_1} + |x_2|^{\alpha_2}$ and a generalized homogeneous angular factor $L(x)$ (Surgailis, 2023).

Under rescaling, partial sums are taken over rectangles with side lengths scaling as $\lambda$ in one axis and $\lambda^\gamma$ in the other. The interplay between the two axes' singularities, controlled by the critical balance exponent $\gamma_0 = \alpha_1 / \alpha_2$, induces fundamentally different limiting fields for subcritical ($\gamma < \gamma_0$), supercritical ($\gamma > \gamma_0$), and critical ($\gamma = \gamma_0$) scaling ratios.

2. Scaling Transition and Classification of Limiting Fields

The "scaling transition" describes the emergence of distinct scaling limits for different scaling exponents. For generic linear fields with anisotropic LRD, the limiting random field is:

• A Fractional Brownian Sheet (FBS) $B_{H_1, H_2}$ with explicitly parameterized Hurst indices away from balance, i.e., for $\gamma \ne \gamma_0$.
• A genuinely two-dimensional, operator-scaling Gaussian field at the critical balance $\gamma = \gamma_0$, which cannot be written as a simple product of one-dimensional processes and is not an FBS.

The functional form of the FBS limit is given by

$$\frac{S_{\lambda, \gamma}(x)}{\lambda^{H(\gamma)}} \Longrightarrow_{\rm fdd} B_{H_1(\gamma), H_2(\gamma)}(x),$$

with Hurst indices expressed as piecewise functions of the spectral exponents and scaling ratio.

At critical balance, the limit becomes

$$V_{\gamma_0}(x) = \int_{\mathbb{R}^2} \prod_{i=1}^2 \frac{1 - e^{\mathrm{i} u_i x_i}}{u_i} \sqrt{f_0(u)}\,Z(\mathrm{d}u),$$

where $Z$ is complex white noise and $f_0$ the leading homogeneous part of the spectrum (Surgailis, 2023).

3. Formal Mechanism and Generalizations

The mechanism underlying scale-dependent transformations derives from competing singularities—typically, non-equivalent scaling exponents—resulting in different axes dominating the joint variance or covariance structures in unbalanced regimes. When the rescaling matches the anisotropy, the contributions from all axes become commensurate, producing a maximally non-degenerate limit; otherwise, the limit degenerates, factorizing into products or displaying independence/invariance along some axes (Puplinskaite et al., 2014, Biermé et al., 2015, Pilipauskaitė et al., 2016).

This paradigm holds in semiparametric models as well. For random fields with general dependence axes (not aligned with the coordinate axes), the scaling transition is determined not by the ratio of spectral exponents, but universally at $\gamma_0=1$. Only isotropic scaling ($\lambda$ in both directions) simultaneously tracks the slowest direction of dependence (Pilipauskaitė et al., 2020).

4. Scale-Dependent Transformations Beyond Linear Fields

Several major generalizations include:

• Nonlinear Functionals: For polynomial functionals (Appell/Hermite chaos) of linear fields, analogous scaling transitions persist, leading to non-Gaussian operator-scaling limits or "Hermite slides" of lower-dimensional processes in unbalanced regimes (Pilipauskaitė et al., 2016).
• Negative Dependence: Even in fields with negative dependence (total coefficient sum zero), scaling transitions manifest, but the limiting fields can become edge-dominated (limits supported predominantly on the boundary of the summation region) with critical values set by integrability criteria on edge exponents. This leads to a richer taxonomy including interior-driven, edge-driven, and mixed Gaussian limits (Surgailis, 2019).
• Non-Gaussian and Infinite Variance Fields: Scaling transitions occur for stable fields, provided their filter structure encodes at least two memory exponents. The presence of a single scaling exponent precludes transitions, as ordinal scaling always yields classical Lamperti-type limits (Damarackas et al., 2019).
• Multifractal Processes: In Multifractal Random Walk (MRW) constructions, scale-dependent field transformations are realized via fractional integration of non-Gaussian fluctuations, and the scaling spectrum

$\zeta_q = qH - \frac{\lambda^2}{2}q(q-1)$

encodes multifractality as a continuous deformation between monofractal and intermittent regimes (Lakhal et al., 2023).

5. Scale-Dependent Field Transformation in Physical and Geometric Contexts

Analogous phenomena arise in field theories with variable coupling parameters. In Asymptotic Safety cosmology and other covariant RG frameworks, the field transformation consists of promoting coupling constants (e.g., $G,\Lambda$) to spacetime fields, introducing kinetic and mixing terms determined uniquely by diffeomorphism invariance and second-order dynamics:

$$G_{\mu\nu} = -\bar\Lambda e^{\psi}g_{\mu\nu} - \frac{1}{2}\psi_{;\mu}\psi_{;\nu} - \frac{1}{4}g_{\mu\nu}\psi^{;\rho}\psi_{;\rho} + \psi_{;\mu;\nu} - g_{\mu\nu}\Box\psi + 8\pi G T_{\mu\nu}$$

$$\left(G T_{\mu\nu}\right)^{;\mu} + G\left(T_{\mu\nu} - \frac{1}{2}Tg_{\mu\nu}\right)\psi^{;\mu}=0$$

Such scale-dependent redefinitions yield new cosmological solutions, including bouncing, recollapsing, or non-singular universes, and directly link the RG scale to spacetime coordinates (Bonanno et al., 2020).

Similarly, in multifield inflation, conformal transformations acting on the Jordan frame action induce a field-space metric whose conformal properties can be systematically classified via scale-dependent analysis of the kinetic sector (Tang et al., 2021).

6. Transformation Methods in Homogenization and PDE Analysis

Scale-dependent field transformations also serve as central tools in mathematical homogenization, particularly in the two-scale transformation method. Here, domain and gradient transformations at the microscale are rigorously mapped to periodic substitute problems using unfolding operators and back-transformation schemes satisfying strong two-scale convergence. The method ensures that effective macroscopic equations are independent of the path taken via local periodic transformations, providing a robust analytical framework for multiscale PDEs (Wiedemann, 2021).

7. Summary Table: Canonical Models and Limiting Behavior

Field Class	Scaling Parameter	Critical Scaling	Limiting Field (Balanced)	Limiting Field (Unbalanced)
Anisotropic Gaussian RF	$\gamma$	$\gamma_0$	Operator-scaling Gaussian	Fractional Brownian Sheet
Nonlinear (Appell) RF	$\gamma$	$\gamma_0$	Non-Gaussian k-th chaos	Hermite slide, FBS, or degenerate
Oblique-dep. LRD field	$\gamma$	$1$	Operator-scaling/well-balanced RF	(Degenerate) FBS with indices
Multifractional (MRW) field	$H$, $\lambda^2$	None (continuum)	MRW with quadratic spectrum	Monofractal regime for large-scale
Negative dependent RF	$\gamma$	$\gamma_0$	Edge- or interior-driven Gaussian	Edge process or FBS, per regime
Scalar field/gravity models	RG or Weyl scale	Model-specific	Einstein-frame, conformal metric	Runaway or singular field behavior

This table links the broad classes of scale-dependent field transformations to their scaling parameters, critical regimes, and limiting processes, as developed in the cited literature (Surgailis, 2023, Puplinskaite et al., 2014, Pilipauskaitė et al., 2020, Lakhal et al., 2023, Bonanno et al., 2020, Wiedemann, 2021, Pilipauskaitė et al., 2016, Damarackas et al., 2019, Surgailis, 2019).

Conclusion

Scale-dependent field transformation is a fundamental structural phenomenon whereby the statistical, dynamical, or geometric nature of a field changes as a function of the interplay between scaling rates and anisotropy. The theory encompasses random and deterministic fields, nonlinear functionals, multifractal processes, and physical field theories. Its rigorous development—especially the classification of limiting fields, identification of scaling transitions, and explicit construction of anisotropic, operator-scaling limits—provides both the main theoretical tools for modern multiscale modeling and powerful connections to statistical mechanics, stochastic analysis, and fundamental physics (Surgailis, 2023, Puplinskaite et al., 2014, Lakhal et al., 2023, Pilipauskaitė et al., 2020, Biermé et al., 2015, Wiedemann, 2021).