Universality of Local Scaling Limits

• The topic describes universality as the emergence of identical local scaling limits in microscopic observables of Lévy-driven fractional random fields under anisotropic rescaling.
• It utilizes increment analysis over ellipsoidal regions to reveal distinct unbalanced and balanced regimes where only key exponents influence the limiting behavior.
• The universality mechanism shows that, except at the critical balanced point, limit fields rely solely on the active direction’s scaling properties, rendering finer details irrelevant.

Universality of local scaling limits refers to the emergence of limit laws or kernel structures for microscopic observables (correlation functions, metric structures, distributions) in diverse probabilistic, combinatorial, or statistical mechanical models after suitable rescaling. These limits display the same functional forms or parameter dependence regardless of fine details of the underlying models, provided coarse features (such as symmetry, decay exponents, or growth dimension) coincide. Universality manifests in random fields, random matrices, geometric structures, scaling limits of combinatorial objects, and dynamical systems far from equilibrium. For Lévy-driven fractional random fields on $\mathbb{R}^2$, as well as related critical phenomena, universality occurs in local small-scale or mesoscopic scaling regimes, with limiting behavior governed by a small set of scaling exponents, symmetry classes, and algebraic relations, and, crucially, is independent of irrelevant microscopic parameters (Pilipauskaitė et al., 2021).

1. Lévy-driven Fractional Random Fields: Model and Scaling Regimes

Consider a random field $X$ on $\mathbb{R}^2$ defined by the moving-average form

$$X(t) = \int_{\mathbb{R}^2} g(t-u)\, M(du), \quad t=(t_1, t_2)$$

where $M$ is an infinitely divisible Lévy random measure in the domain of attraction of an $\alpha$-stable sheet, $0<\alpha\leq2$. The kernel $g$ obeys near the origin a power law: $$g(t) \sim \left(|t_1|^{q_1} + |t_2|^{q_2}\right)^\chi\, L(t)$$ with exponents $q_i>0$, parameter $\chi\neq 0$, and $L(t)$ bounded.

To analyze local behavior, consider increments over ellipsoidal regions shrinking as $\lambda\to0$:

• Ordinary increment: $\Delta_{h,k}X(x) = X(x_1+h, x_2+k) - X(x_1, x_2)$
• Rectangular increment: $$\Box_{h,k}X(x) = X(x_1+h, x_2+k) - X(x_1, x_2+k) - X(x_1+h, x_2) + X(x_1, x_2)$$

Scaling is anisotropic: set $(\lambda, \lambda^\gamma)$ with $\gamma>0$ for horizontal/vertical directions. Limits of rescaled increments,

$$T_\gamma(t) := \lim_{\lambda\rightarrow 0} \lambda^{-H(\gamma)} \Delta_{(\lambda,\lambda^\gamma)} X(t), \qquad R_\gamma(t) := \lim_{\lambda\rightarrow 0}\lambda^{-H(\gamma)} \Box_{(\lambda,\lambda^\gamma)} X(t)$$

are sought for $\gamma>0$, with the scaling exponent $H(\gamma)$ chosen non-degenerate.

2. Existence of Local Anisotropic Scaling Limits and Critical Anisotropy

A critical parameter, $\gamma_0 = q_1/q_2 = p_1/p_2$, termed the intrinsic dependence ratio, demarcates three scaling regimes:

• For $\gamma < \gamma_0$: limits $T_\gamma\equiv T_-$, $R_\gamma\equiv R_-$, independent of $\gamma$.
• For $\gamma > \gamma_0$: limits $T_\gamma\equiv T_+$, $R_\gamma\equiv R_+$, independent of $\gamma$.
• For $\gamma = \gamma_0$: distinct balanced limits $T_0, R_0$, retaining the full anisotropic structure.

All $\gamma>0$ yield well-defined $\alpha$-stable limit fields under mild regularity and tail assumptions on $g$ and $M$ (Pilipauskaitė et al., 2021).

3. Structure of Unbalanced and Balanced Local Limits

Unbalanced limit fields for both increment types have sharp structure:

• Rectangular (second-order) increments (MSS fields): For $R_-$ and $R_+$, the limiting field is multi-self-similar (MSS) with Hurst indices $(H_1,H_2)$, one being $0$ or $1$. For instance, if $\gamma<\gamma_0$ and $P_{1/\alpha,(1+\alpha)/\alpha}<1$, $R_- = \Upsilon_{\alpha,1}$ is $(H_{\alpha,1},0)$-MSS; formulas for exponents are explicit. In the Gaussian case $\alpha=2$, these correspond to fractional Brownian sheets $B_{H_1,H_2}$ with one exponent at $0$ or $1$.
• Ordinary (first-order) increments (tangent limits): $T_-$ and $T_+$ depend only on one coordinate, e.g. for $\gamma<\gamma_0$, $T_-(t_1,t_2)=T_0(t_2)$, an $\alpha$-stable process in the vertical direction only.

For balanced case ($\gamma = \gamma_0$), both exponents $p_1,p_2$ enter, yielding a genuine two-dimensional self-similar, operator-scaling field.

4. Universality Mechanism and Parameter Collapse

The universality phenomenon for these fields is characterized by remarkable collapse:

• Active-direction universality: In each unbalanced regime ($\gamma<\gamma_0$ or $\gamma>\gamma_0$), the limiting field (e.g. $R_-$) depends only on the stability index $\alpha$ and one exponent ($p_1$ or $p_2$), not on $\gamma$ or on the other exponents.
• Balanced regime retention: Only at the critical $\gamma_0$ does the limit depend on both exponents.
• Transition and independence: All choices of $\gamma<\gamma_0$ yield the same $R_-$, and similarly for $\gamma>\gamma_0$; the scaling limits are strictly independent of the anisotropy parameter away from the critical point.

This demonstrates a universality class indexed by $(\alpha, p_1)$ or $(\alpha, p_2)$, with the field’s multi-self-similar structure entirely determined by the “active” direction’s scaling properties.

5. Connections to Large-scale Anisotropic Scaling and Prior Results

The scaling transition and universality ranks among the most robust phenomena for Lévy-driven fractional fields:

• Large-scale analogs: The large-scale ($\lambda\to\infty$) anisotropic scaling of random fields on $\mathbb{Z}^2$ exhibits similar trichotomy, with unbalanced Gaussian/stable sheets possessing anomalous or degenerate Hurst indices [Pilipauskaitė & Surgailis 2017].
• Isotropic case: For $q_1=q_2$, the balanced regime reproduces the classical 1-tangent limit for isotropic Lévy-fractional fields [Benassi, Cohen & Istas 2004].

These scaling trichotomies and limit structures arise robustly across spatial models and scales, including integer-lattice models and random fabrics (Pilipauskaitė et al., 2021).

6. Summary and Universality Principle

For Lévy-driven fractional random fields with moving-average kernels of the generic power-law form,

$g(t) \sim |t_1|^{-p_1} + |t_2|^{-p_2}$

the local anisotropic scaling limits under arbitrarily thin rectangles collapse, except at balance, to universal limit fields indexed solely by the minimal (or maximal) decay exponent. The critical (balanced) regime preserves full anisotropic information. This principle exemplifies broad universality: scaling limits and local geometric structure are determined by a minimal set of parameters, with all finer details of anisotropy or kernel regularity rendered irrelevant except at isolated transition points (Pilipauskaitė et al., 2021).