Oberseminar Analysis
A talk in the Oberseminar Analysis series by
Benoit Dagallier from U Paris Dauphine
| Abstract: | We consider particles interacting with a smooth mean-field potential and attempt to quantify the speed of convergence (log-Sobolev constant) of the associated Langevin dynamics in terms of the number N of particles and the strength of the interaction. Our main interest is in relating the scaling of the log-Sobolev constant as a function of N, to properties of the free energy of the model. We show that a certain notion of convexity of the free energy implies uniform-in-N bounds on the log-Sobolev constant. In some cases this convexity criterion is sharp, for instance this is the case in the Curie-Weiss model where we prove uniform bounds on the log-Sobolev constant up to the critical temperature, which is optimal. Our proof does not involve the dynamics. Instead, we decompose the measure describing interactions between particles with inspiration from renormalisation group arguments for lattice models, that we adapt here to a lattice-free setting in the simplest case of mean-field interactions. Our results apply more generally to non mean-field, possibly random settings, provided each particle interacts with sufficiently many others. Based on joint work with Roland Bauerschmidt and Thierry Bodineau. $\href{https://uni-bielefeld.zoom-x.de/j/62228418411?pwd=bkNteGxqMGFRemZhOWRwek1tNGtkUT09}{\textbf{Join Zoom Meeting}}$ Meeting-ID: 622 2841 8411 Passwort: 058508 Within the CRC this talk is associated to the project(s): A8 |