Log-Sobolev inequality for the continuum phi^4_2 and phi^4_3 models

26.09.2022 16:15 – 18:15

In this talk, I will report on a joint work with Roland Bauerschmidt in which we analyse the Langevin dynamics associated with the continuum phi^4 model. The continuum phi^4 model is one of the simplest models of field theory, introduced at least 50 years ago in the physics community. It can be thought of as a continuous analogue of the Ising model. In particular, it exhibits a phase transition, separating a weakly correlated and a strongly correlated regime. The presence of a phase transition should imply a dramatic difference in how much time it takes for the dynamics to approach its steady state, from fast convergence above the critical point, to slow convergence diverging with the system size below it. The analysis of the model is made particularly subtle due to the fact that the continuum limit is ill-defined, in the sense that a certain renormalisation procedure is needed to make sense of it. We characterise the relaxation to the steady state by proving a logarithmic Sobolev inequality (LSI) with constant bounded under optimal assumptions. The proof makes use of a very general LSI criterion developed in 2019 by Roland Bauerschmidt and Thierry Bodineau. In the talk, I will introduce the phi^4 model and LSI inequalities, then try to explain the main ideas as non-technically as possible.

Lieu

Salle 1-15, Séminaire "Maths-Physique"

entrée libre