From smooth to discontinuous kernels: a variance transfer principle for hyperuniform processes
Abstract
We give a transfer principle for the fluctuations of linear statistics of finite particle systems around Lebesgue measure: if for a smooth kernel the variance decays polynomially with some exponent a compared to independent (non-interacting) particles, then the number variance over balls centred at almost every point decays with exponent min(1,a) times a log term if a=1, over a possibly reduced range of scales for non-periodic systems. We apply this principle to eigenvalues of random N*N Girko matrices, leveraging results of Cipolloni et al., and obtain the optimal perimeter-like number variance, on the microscopic and some mesoscopic scales range, after local averaging. We also apply the results to Coulomb gases, by transferring the results of Serfaty: we prove that 2D-Coulomb gases are 2-hyperuniform, i.e. they have surface order number variance, which is optimal, and that 3D Coulomb gases are 1-hyperuniform. In 2D, it allows to prove finite Coulomb energy and Wasserstein distance to Lebesgue measure.
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BibTeX
@misc{lachiezerey2026from,
title = {{From smooth to discontinuous kernels: a variance transfer principle for hyperuniform processes}},
author = {Raphaël Lachièze-Rey},
howpublished = {HAL},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.22940},
eprint = {2606.22940},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05663246v3},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - From smooth to discontinuous kernels: a variance transfer principle for hyperuniform processes
AU - Raphaël Lachièze-Rey
PY - 2026
DA - 2026/06/19
PB - HAL
DO - 10.48550/arxiv.2606.22940
UR - https://hal.science/hal-05663246v3
AB - We give a transfer principle for the fluctuations of linear statistics of finite particle systems around Lebesgue measure: if for a smooth kernel the variance decays polynomially with some exponent a compared to independent (non-interacting) particles, then the number variance over balls centred at almost every point decays with exponent min(1,a) times a log term if a=1, over a possibly reduced range of scales for non-periodic systems. We apply this principle to eigenvalues of random N*N Girko matrices, leveraging results of Cipolloni et al., and obtain the optimal perimeter-like number variance, on the microscopic and some mesoscopic scales range, after local averaging. We also apply the results to Coulomb gases, by transferring the results of Serfaty: we prove that 2D-Coulomb gases are 2-hyperuniform, i.e. they have surface order number variance, which is optimal, and that 3D Coulomb gases are 1-hyperuniform. In 2D, it allows to prove finite Coulomb energy and Wasserstein distance to Lebesgue measure.
ER -