In the first talk I will introduce the Polchinski flow (or dynamics), a general framework to study asymptotic properties of statistical mechanics and field theory models, inspired by renormalisation group ideas. The Polchinski dynamics has appeared recently under different names, such as stochastic localisation, and in very different contexts. Here I will motivate its construction in detail from a physics point of view and mention a few applications. I will in particular explain how the Polchinski flow can be used to generalise Bakry and Emery’s Γ2 calculus to obtain functional inequalities (Poincaré, log-Sobolev...).
This first part is based on a review paper with Roland Bauerschmidt and Thierry Bodineau, accessible here: https://arxiv.org/pdf/2307.07619
In the second part I will focus on the \Phi^4_2 singular SPDE in infinite volume. The goal is to prove uniqueness of its invariant measure. In finite volume this is well known due to works of Tsatsoulis and Weber. In infinite volume, however, phase transitions, static or dynamical, suggest the existence of either one or two invariant measures depending on the parameters of the model. For lattice models, Holley, Stroock and Zegarlinski in the 80s and 90s have proposed a general roadmap on proving uniqueness of infinite volume invariant measures using log-Sobolev inequalities. I will explain how to generalise these ideas in the continuum in the \Phi^4_2 case.
This second part is based on ongoing work with Roland Bauerschmidt and Hendrik Weber.
