MARK KAC SEMINAR

Mesoscopic fluctuations of microscopic (discrete or continuous) dynamics can be described in terms of non-Gaussian random fields. These random fields are fully described by certain nonlinear stochastic partial differential equations which are universal: they depend on very few details of the microscopic model. However, due to the extreme irregular nature of the random field sample paths, these equations turn out to not be well-posed in any classical analytic sense. In this talk I will review recent progress in the mathematical understanding of such singular equations and of their (weak) universality. In particular I will discuss the case of the one dimensional Kardar–Parisi–Zhang equation and of the three dimensional Stochasic Allen–Cahn equation.

A Gibbs-non-Gibbs transition is a transition from a state that is Gibbsian into a non-Gibbsian one by a transformation. For example, a Gibbs measure for the Ising model transformed by certain spin-flip dynamics may become non-Gibbs. Understanding Gibbs-non-Gibbs transitions on the “level” of lattices or graphs on which Gibbs measures are defined is rather difficult as one has to deal with the underlying graph structure.

In this talk I will describe mean-field systems and their Gibbs-non-Gibbs transitions. Mean-field systems do not have an underlying graph structure, as the interaction is equally distributed among all spins; the interaction is “mean-field”. In most of the considered mean-field systems the conjecture about the relation between minimisers of a rate function and Gibbsianness has been proved. This relation and consequences in different examples will be described. Possibly, I will also go into level 2 mean-field systems.