A criterion on the free energy for log-Sobolev inequalities in mean-field particle systems

Abstract: For a class of mean-field particle systems, we formulate a criterion in terms of the free energy that implies uniform bounds on the log-Sobolev constant of the associated Langevin dynamics. For certain double-well potentials with quadratic interaction, the criterion holds up to the critical temperature of the model, and we also obtain precise asymptotics on the decay of the log-Sobolev constant when approaching the critical point. The criterion also applies to ``diluted'' mean-field models defined on sufficiently dense, possibly random graphs. We further generalize the criterion to non-quadratic interactions that admit a mode decomposition. The mode decomposition is different from the scale decomposition of the Polchinski flow we used for short-range spin systems.