May 13, 2011
Location: Janskerkhof 15a (Utrecht), room 001
Some recent results on freezing Gibbs measures
We report on recent positive and negative results for the convergence of Gibbs measures as temperature goes to zero. The Gibbs measures considered are on {0,1}Zd.
Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model
For two-dimensional independent percolation, Russo–Seymour–Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be a key tool to describe the phase transition: what happens at and near criticality.
In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes some harmonicity appear on the discrete level, providing precise estimates on boundary connection probabilities.
We also prove several related results — including some new ones — among which are the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent.
This is joint work with H. Duminil-Copin and C. Hongler.