Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime
Abstract
We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert--Schmidt regime, we prove N-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman--Fredholm determinant of the centered interaction operator. We show that the Hilbert--Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two U-statistics. As consequences, we establish entropic commutator estimates with the sharp O(N−1) additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert--Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.
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BibTeX
@misc{delgadino2026sharp,
title = {{Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime}},
author = {Matias G. Delgadino and Rishabh Gvalani and Matthew Rosenzweig},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02743},
eprint = {2609.02743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02743},
}
RIS
TY - UNPB
TI - Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime
AU - Matias G. Delgadino
AU - Rishabh Gvalani
AU - Matthew Rosenzweig
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02743
UR - https://arxiv.org/abs/2609.02743
AB - We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert--Schmidt regime, we prove N-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman--Fredholm determinant of the centered interaction operator. We show that the Hilbert--Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two U-statistics. As consequences, we establish entropic commutator estimates with the sharp O(N^−1) additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert--Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.
ER -