Gaussian fluctuations of differential observables of stationary lattice fields
Abstract
We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete d-dimensional torus of mesh 1/N. Our main result is a Central Limit Theorem for the fields f↦ N−d/2−m∑xϕxN(Pf)(x/N), where P is a scalar homogeneous constant-coefficient differential operator of order m≥0 with Fourier symbol p, e.g., P=Δ and m=2. Under suitable assumptions on the spectral scaling limit γ of the covariance and the normalized cumulants of order r≥3, we prove convergence in law in a negative Sobolev space to a centered Gaussian random distribution with covariance multiplier |p(k)|2γ(k). The two factors separate the effects of the operator and the field and, notably, differentiation can compensate a low-frequency spectral singularity. For the lattice Gaussian Free Field, this gives a projected white-noise limit for the gradient. The proofs use Fourier analysis and the method of moments and cumulants, together with a tightness argument. Finally, we give two sufficient criteria for the cumulant assumption, namely a tree--graph bound and the Dobrushin uniqueness condition for finite-range Gibbs fields, and recover the white-noise limits of the centered Ising magnetization and Potts colour fluctuation fields at high temperature.
Related
- Dynamical and stationary non-equilibrium fluctuations for a boundary-driven exclusion process with energy
- Fluctuations of Nonlinear Observables in Mean Field Neural Network Training
- Threshold Tails of Black hole Differential Observables
- Spectral Invariance for Black-Hole Differential Observables
- Hamiltonian stationary Lagrangian surfaces of constant Gaussian curvature in complex space forms
BibTeX
@misc{coppini2026gaussian,
title = {{Gaussian fluctuations of differential observables of stationary lattice fields}},
author = {Fabio Coppini and Wioletta M. Ruszel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04452},
eprint = {2610.04452},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04452},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian fluctuations of differential observables of stationary lattice fields
AU - Fabio Coppini
AU - Wioletta M. Ruszel
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04452
UR - https://arxiv.org/abs/2610.04452
AB - We study Gaussian fluctuations of differential observables of centered stationary random fields on the discrete d-dimensional torus of mesh 1/N. Our main result is a Central Limit Theorem for the fields f↦ N^−d/2−m∑_xϕ_x^N(Pf)(x/N), where P is a scalar homogeneous constant-coefficient differential operator of order m≥0 with Fourier symbol p, e.g., P=Δ and m=2. Under suitable assumptions on the spectral scaling limit γ of the covariance and the normalized cumulants of order r≥3, we prove convergence in law in a negative Sobolev space to a centered Gaussian random distribution with covariance multiplier |p(k)|^2γ(k). The two factors separate the effects of the operator and the field and, notably, differentiation can compensate a low-frequency spectral singularity. For the lattice Gaussian Free Field, this gives a projected white-noise limit for the gradient. The proofs use Fourier analysis and the method of moments and cumulants, together with a tightness argument. Finally, we give two sufficient criteria for the cumulant assumption, namely a tree--graph bound and the Dobrushin uniqueness condition for finite-range Gibbs fields, and recover the white-noise limits of the centered Ising magnetization and Potts colour fluctuation fields at high temperature.
ER -