A criterion on the free energy for log-Sobolev inequalities in mean-field particle systems
R. Bauerschmidt, T. Bodineau, B. Dagallier
TL;DR
The paper develops a renormalised, mode-based framework to bound the log-Sobolev constant for mean-field Langevin dynamics across both quadratic and non-quadratic interactions. It links uniform relaxation to macroscopic free-energy properties, notably through a Polyak–Łojasiewicz criterion for a coarse-grained free energy $\hat{\mathcal{F}}_T$ and a strongly convex renormalised potential $\mathcal{V}_T$. In the quadratic case, it proves uniform LSI up to the critical temperature $T_c$, with precise near-critical scaling, and extends to random/diluted graphs, where graph spectrum governs the bounds. For general interactions, a mode-decomposition approach yields a practical criterion ensuring $N$-uniform LSI, unifying analysis for fully connected and graph-based mean-field models via renormalisation and Bakry–Émery-like arguments.
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.