Near-critical Ising mode.

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