The near-critical planar Ising Random Cluster model
A cornerstone of statistical physics is the determination of the critical temperature and some critical exponents of the planar Ising spin model by Onsager (1944). In particular, these give the correlation length exponent of the Ising Random Cluster model (a dependent bond percolation model describing the correlations between the Ising spins), which determines how fast a large finite system becomes supercritical from subcritical as the temperature is lowered.
However, Onsager's approach is very non-geometric and somewhat mysterious for mathematicians. In this talk, I will discuss a definition of the correlation length via crossing probabilities, and its computation using Smirnov's fermionic observable. Secondly, I will highlight a striking phenomenon about the near-critical behavior of the Ising Random Cluster model, which is completely missing from the case of standard percolation: in any monotone coupling of the Random Cluster configurations \omega_p (e.g., in the one introduced by Grimmett 1995), as one raises p near p_c, the new edges arrive in a fascinating self-organized way, so that the correlation length is not governed anymore by the amount of pivotal edges at criticality.
This is joint work with Hugo Duminil-Copin and Christophe Garban.