From one particle to N particles: uniform transport-information and log-Sobolev inequalities for Coulomb gases on the torus
Abstract
We study how a functional inequality satisfied by the one-particle conditional distributions of a Gibbs measure, with a constant independent of the positions of the other particles, can be propagated to the whole N -particle system, with a constant independent of N , when the interaction is singular. Our model is the Coulomb gas on the torus T 3 in the mean-field regime. We distinguish two mechanisms. The first one is a dependent tensorization, governed by Dobrushin-Wu interdependence coefficients. It yields uniform Poincar{é} and transport-information W 2 I inequalities with fully explicit constants. We explain why this mechanism does not extend to the log-Sobolev inequality: the Zegarlinski-type coefficients, which require pointwise bounds on the interaction, are infinite. The second mechanism is a tensorization with a defect. Using a uniform bound on the partition function due to Duerinckx and Jabin, it yields a uniform log-Sobolev inequality for the Coulomb gas on T 3 at small charge. We identify the assumptions on the interaction under which each mechanism applies. All the results are then extended to the two-dimensional Coulomb gas and to periodic Riesz gases 1/|x|^s , 0 < s ≤ 1. The extension to d ≥ 4 is also discussed.
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BibTeX
@misc{guillin2026from,
title = {{From one particle to N particles: uniform transport-information and log-Sobolev inequalities for Coulomb gases on the torus}},
author = {A Guillin and W Liu and B Nectoux and L Wu},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35350},
eprint = {2610.35350},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05839088v2},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - From one particle to N particles: uniform transport-information and log-Sobolev inequalities for Coulomb gases on the torus
AU - A Guillin
AU - W Liu
AU - B Nectoux
AU - L Wu
PY - 2026
DA - 2026/10/26
PB - HAL
DO - 10.48550/arxiv.2610.35350
UR - https://hal.science/hal-05839088v2
AB - We study how a functional inequality satisfied by the one-particle conditional distributions of a Gibbs measure, with a constant independent of the positions of the other particles, can be propagated to the whole N -particle system, with a constant independent of N , when the interaction is singular. Our model is the Coulomb gas on the torus T 3 in the mean-field regime. We distinguish two mechanisms. The first one is a dependent tensorization, governed by Dobrushin-Wu interdependence coefficients. It yields uniform Poincar{é} and transport-information W 2 I inequalities with fully explicit constants. We explain why this mechanism does not extend to the log-Sobolev inequality: the Zegarlinski-type coefficients, which require pointwise bounds on the interaction, are infinite. The second mechanism is a tensorization with a defect. Using a uniform bound on the partition function due to Duerinckx and Jabin, it yields a uniform log-Sobolev inequality for the Coulomb gas on T 3 at small charge. We identify the assumptions on the interaction under which each mechanism applies. All the results are then extended to the two-dimensional Coulomb gas and to periodic Riesz gases 1/|x|^s , 0 < s ≤ 1. The extension to d ≥ 4 is also discussed.
ER -