The \Phi^4 model: Renormalisation, particle approximations and global analysis"

11.11.2015, 16:15 - 17:45  –  TU Berlin room MA748 (RTG Lounge)
Gastvortrag

Hendrik Weber (Warwick)

The \Phi^4 model is a classical model in statistical mechanics. It arises naturally when trying to

construct a version of the Ising model with continuous spins (rather than +1 0r -1 valued spins) dened on

continuous space (rather than a lattice). The construction of this model involves dicult issues such as

working with random distributions rather than functions and the appearance of "innite terms" that

have to be removed. Over the last few years remarkable progress has been made in the understanding of

such singular stochastic objects - most prominently by Hairer and Gubinelli - and in this series of lectu-

res I aim to present some of the key ideas and their implications in the context of this very interesting model.

The rst lecture will be introductory: I will explain in some detail the derivation of the 4 model, basic

scaling and regularity issues as well the concept of renormalisation. I will then show a perturbative

approach which is at the heart of the solution theories by Hairer and Gubinelli. The main aim of the

second lecture will be to discuss how the 4 model arises as a scaling limit of Ising models with long

range interaction. Finally, the third lecture I will show how to obtain bounds on behaviour of the system

at large scales with PDE methods. The results presented in the second and third lecture are joint work

with Jean-Christophe Mourrat.