Statistical physics, renormalisation, and Log-Sobolev inequalities

Statistical physics is concerned with the understanding of the large scale behaviour of microscopically defined models. Simple representative models that have been studied starting 100 years ago, and continue to be intensively studied today, include the Ising and Heisenberg models, Percolation, Interface models, Random Schroedinger Operators, Random Polymers, Random Walks in Random Environments, and many others.

As a function of a temperature parameter (or similar), these models often undergo phase transition that connect qualitatively very different behaviour. Understanding the nature of these transitions, and often related universal behaviour that is independent of precise microscopic definitions, remain fundamental mathematical questions. After giving a short overview of the status, I will highlight the concept of renormalisation that plays a fundamental role in physics in explaining many of these questions at a heuristic level.

In the second part of the talk, I will give a concrete example of the use of renormalisation in understanding stochastic dynamics of such models, an aspect that is mathematically much less understood than its static counterpart. We show that the idea of the renormalisation group in the spirit of Polchinski connects naturally to the classical theory for Log-Sobolev inequalities of Bakry and Emery. As an application, we prove a Log-Sobolev inequality for the continuum Sine-Gordon model.

(The second part is joint work with T. Bodineau.)