From one particle to N particles: uniform transport-information and log-Sobolev inequalities for Coulomb gases on the torus
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 $\le$ 1. The extension to d $\ge$ 4 is also discussed.