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.
From: Raphael Lachieze-Rey [view email] [via CCSD proxy]
[v1]
Mon, 22 Jun 2026 07:18:00 UTC (45 KB)
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