
Conference: Symposium on Mathematical Physics
Lattice models: criticality, universality and other questions
Prof. Dr. Ioan Manolescu's talk Date: 11.11.14 Time: 09.00  10.00 Room: Y27H28
The related notions of phase transition and universality, essential in
physics, have, in the last halfcentury, penetrated and established
themselves in mathematics, for instance through the probabilistic
study of lattice models. One such model is the randomcluster model,
of which percolation is a particular instance. It is a model of random
connections in a graph. When considered on an infinite regular graph
(or lattice), it exhibits a phase transition expressed in terms of the
existence of infinite connected components. In this talk I aim to
present recent results and open questions revolving around how this
phase transition occurs on ddimensional lattices, for instance on
Z^d.
We will start with the simplest version of the model: planar
percolation. In this particular setting much progress has been made,
and recently we have obtained the first evidence of universality. We
will then move on to the planar randomcluster model, where additional
difficulties appear and the phase transition is more complex. In this
case we will discuss several new results and many open questions. To
finish we will mention some directions of research in the much more
obscure field of randomcluster models in dimensions d > 2.
If time allows, I will explain the link between the randomcluster
model and other statistical mechanics models, namely the Potts model
and selfavoiding random walk.

