Near-critical Ising mode.

28 January 2013
In this talk, I will present two results on the behavior of the Ising model on the planar lattice near its critical point: (i) In the first result (joint work with F.Camia and C. Newman), we will fix the temperature to be the critical temperature T_c and we will vary the magnetic field h \geq 0. Our main result states that in the plane Z^2, the average magnetization at the origin behaves up to constants like h^{1/15}. This result is interesting since the classical computa- tion of the average magnetization by Onsager requires the external magnetic field h to be exactly 0 . (ii) In the second result (joint work with H. Duminil-Copin and G. Pete), we focus on the correlation length of the Ising model when h is now fixed to be zero and one varies instead the temperature T around T_c. In rough terms, if T<T_c, the correlation length L(T) is the scale above which things start looking supercritical and below which the system still “looks” critical. We prove that (up to log corrections), L(T) is of order |T-T_c|^{-1}. It turns out that the study of the associated near-critical FK model reveals a very interesting self-organization type of structure.
  • Stochastic Analysis Seminar