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.
