Discontinuity of the phase transition for the planar random-cluster and Potts models with q > 4

The ferromagnetic q-Potts Model is a classical spin system in which one of q colors is placed at every vertex of a graph and assigned an energy proportional to the number of monochromatic neighbors. It is highly related to the Random Cluster model, which is a dependent percolation model where a configuration is weighted by q to the power of the number of clusters. Through non-rigorous means, Baxter showed that the phase transition is first-order whenever q > 4 – i.e. there exists multiple Gibbs states at criticality. We provide a rigorous proof of the second claim. Like Baxter, our proof uses the correspondence between the above models and the Six-Vertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter’s formula for the correlation length of the models at criticality. This is joint work with Hugo Duminil-Copin, Maxemine Gangebin, Ioan Manolescu, and Vincent Tassion.

• Speaker: Matan Harel (IHES Paris)
• Tuesday 25 October 2016, 16:30–17:30
• Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
• Series: Probability; organiser: Perla Sousi.