MARK KAC SEMINAR

The aim of this talk is to present in some detail our recent results with Hugo Duminil-Copin on the two-dimensional random-cluster model on the two-dimensional square lattice, namely the determination of the critical point and the sharpness of the phase transition.

There are now two proofs of the result, the first one deriving p_c through the use of sharp-threshold results and RSW-like estimates, and the second one exploiting Smirnov’s fermionic observable away from the self-dual point to gain estimates on two-point functions. I will first introduce the general strategy of the first argument in the case of percolation and extend it to the random-cluster model, giving a mostly self-contained proof. If time allows, I will say a few words about the second proof, which is restricted to the case q>4 but extends to a larger family of lattices.

Below the critical temperature, the Gibbs states of the nearest neighbor Ising model on Z² are convex combinations of the two pure phases. In particular, they are all translation invariant, and no phase coexistence occurs. This result generalizes to the q-state Potts model, which has q pure phases in the subcritical regime.

I will first explain the basic concepts underlying these statements and survey the main historical results on these questions. Then, I will present the heuristics of the proofs, which consists of considering the model in large finite boxes with arbitrary boundary condition, and proving that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature β > β_c(q) = log(1+√q).

This is a joint work with Hugo Duminil-Copin, Dima Ioffe and Yvan Velenik.