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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 half-century, penetrated and established themselves in mathematics, for instance through the probabilistic study of lattice models. One such model is the random-cluster 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 d-dimensional 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 random-cluster 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 random-cluster models in dimensions d > 2. If time allows, I will explain the link between the random-cluster model and other statistical mechanics models, namely the Potts model and self-avoiding random walk.