Functions of Bounded Variation and Point Processes

Abstract: We investigate the relationship between the analytical properties of functions of bounded variation and the statistical behavior of hyperuniform point processes. We establish several characterization formulas for the jump part of the gradient of a bounded variation function, extending and unifying previous results by Beretti--Gennaioli and Dávila. In particular, we provide new expressions for the $L<sup>2$-jump of the gradient using both difference quotients and Fourier transform methods. Furthermore, we connect these analytic structures to the theory of hyperuniform point processes. By analyzing the variance of linear statistics associated with bounded variation functions, we provide asymptotic estimates that depend on the specific classification of the hyperuniformity of the point process. The results show how the regularity and jump discontinuities of a function dictate the growth rate of fluctuations in point processes. Finally, we introduce an averaged quadratic BMO-type oscillation functional over translated and rotated cube partitions, similar to the one recently studied by Ambrosio et al., and prove, using results from point process, that it converges to an explicit dimensional constant times the $L<sup>2-$jump, giving in particular a further new characterization of the perimeter of a set.

• The paper establishes a unified framework linking BV functions' analytic properties with variance scaling in hyperuniform point processes.
• It rigorously equates nonlocal difference quotient and Fourier formulations to characterize jump discontinuities and set perimeters.
• Explicit scaling laws and asymptotic formulas are derived, offering practical insights for applications in statistical physics and geometric analysis.

Analytical Frameworks for Functions of Bounded Variation and Hyperuniform Point Processes

Introduction

The paper "Functions of Bounded Variation and Point Processes" (2606.08304) develops a rigorous connection between the analytic properties of functions of bounded variation (BV) and the statistical aspects of hyperuniform point processes. The authors synthesize classical and recent nonlocal characterizations of BV functions, particularly focusing on the jump part of their distributional derivatives, and extend these results to precise asymptotic formulas controlling fluctuations in hyperuniform point processes. The paper provides unified criteria encompassing difference quotients, Fourier-based formulations, and BMO-type oscillation functionals, illustrating how regularity and discontinuities in BV functions influence the scaling regimes in point statistics.

Nonlocal and Fourier Characterizations of BV and Perimeter

The BV space on $\mathbb{R}^d$ includes $L^1$ functions whose distributional gradient $Du$ is a finite vector-valued Radon measure. Classical characterizations, such as those of Bourgain-Brezis-Mironescu and Dávila, express the BV seminorm as limits of nonlocal difference quotients weighted by mollifiers. The paper shows that these formulations, which do not directly depend on distributional derivatives, are equivalent to integral expressions involving Fourier transforms.

Specifically, the authors rigorously connect the Dávila-type difference quotient characterization with recent Fourier criteria from Beretti–Gennaioli. This joint approach establishes:

• For characteristic functions $\chi_\Omega$ of sets $\Omega$ with finite measure, the perimeter $\Per(\Omega)$ can be obtained from asymptotic integrals:

$\lim_{R \to \infty} \frac{2\pi^2}{R} \int_{|\xi| < R} |\xi|^2 |\widehat{\chi_\Omega}(\xi)|^2\, d\xi = \Per(\Omega)$

where $\widehat{\chi_\Omega}$ denotes the Fourier transform with conventions matching Plancherel's theorem.

• The perimeter characterization is improved; the authors demonstrate that bounding the $\liminf$ is sufficient (contradicting previous dependence on $\limsup$), thus relaxing the prerequisite regularity assumptions.
• For general $L^1$0, the jump part of $L^1$1, denoted $L^1$2, is captured via:

$L^1$3

This formula, valid under mild conditions, distinguishes the $L^1$4-jump component from the total variation.

The paper generalizes these spectral formulas across scaling regimes, parameterized by $L^1$5, with sharp asymptotics for $L^1$6, $L^1$7, and $L^1$8. The authors verify that Fourier-based jump norms coincide with difference-quotient-based norms, providing a unified family of formulas, including for fractional Sobolev norms when $L^1$9.

Nonlocal Difference Quotient Formulas for the Jump Part

The authors derive asymptotic difference quotient formulas for the $Du$0 jump, extending the Dávila framework to quadratic forms:

$Du$1

where $Du$2 is a suitable radial mollifier and $Du$3 is determined by the low-frequency behavior of the Fourier transform of $Du$4. Corresponding logarithmic scaling formulas are established for $Du$5. These results strengthen characterizations for sets via their characteristic functions and answer open questions regarding extension from SBV to BV spaces.

Hyperuniform Point Processes: Variance Asymptotics and Linear Statistics

The paper examines random point processes, specifically hyperuniform processes, whose structure function $Du$6 vanishes as $Du$7. Hyperuniformity implies suppressed density fluctuations in large windows and arises across condensed matter and random matrix theory.

The structure function's asymptotic expansion near the origin,

$Du$8

induces three scaling regimes for variance $Du$9 of linear statistics $\chi_\Omega$0:

• Type I ($\chi_\Omega$1): $\chi_\Omega$2, dictated by the jump norm.
• Type II ($\chi_\Omega$3): $\chi_\Omega$4.
• Type III ($\chi_\Omega$5): $\chi_\Omega$6, determined by fractional norms.

For $\chi_\Omega$7, the variance asymptotically depends only on jump discontinuities:

$\chi_\Omega$8

with explicit constants computable from the structure function. For specific point processes, such as the zeros of the Gaussian Entire Function (GEF), the authors derive closed-form coefficients involving the Riemann zeta function $\chi_\Omega$9.

The paper also observes that, for non-BV sets, the variance scaling is controlled by the Minkowski dimension, and for characteristic functions of fractal sets such as the Koch snowflake, the variance formula is finite only for restricted values of $\Omega$0.

BMO-Type Oscillation Functionals and Characterization of Perimeter

Motivated by Ambrosio et al., the authors introduce an averaged quadratic BMO-type functional over translated and rotated cube partitions:

$\Omega$1

averaged over all translations $\Omega$2 and rotations $\Omega$3. Using point process theory, it is proved that as $\Omega$4,

$\Omega$5

with $\Omega$6 an explicit dimensional constant. In particular, for sets $\Omega$7 with finite perimeter,

$\Omega$8

thus providing an alternative operational characterization of the perimeter without relying on distributional derivatives.

Implications and Future Directions

The formal integration of nonlocal, Fourier, and probabilistic frameworks for BV functions and hyperuniform point processes opens avenues for precise control of random fluctuations in statistical physics, materials science, and geometric measure theory. The results question the necessity of classical assumptions, notably relaxing $\Omega$9 conditions in Fourier perimeter characterizations, and suggest richer operational tools for computer-assisted analysis of random processes and geometric boundaries.

The explicit connection between jump discontinuities and variance scaling in hyperuniform systems informs randomized algorithms, robust statistical estimation, and simulation of random geometries. In the context of random matrix theory and determinantal processes, the universality of scaling regimes tied to BV function properties elucidates underlying symmetries and geometric invariants.

Potential extensions include analysis on manifolds, incorporation of anisotropic or non-stationary point processes, and further exploration of norm decompositions relating to the Cantor part of the derivative.

Conclusion

The paper establishes a unified analytic-statistical paradigm linking BV functions and hyperuniform point processes. By proving equivalence and sharpening the criteria among difference quotients, Fourier integrals, and BMO-type oscillation norms, the authors provide a comprehensive framework for understanding singularities, perimeter, and variance in random systems. The strong numerical results, including explicit scaling laws and constants, validate the practical applicability and theoretical significance of these connections in geometric analysis and probability (2606.08304).