We prove that the pressure (or free energy) of the finite range ferromagnetic
Ising model on Zd is analytic as a function of both the inverse
temperature β and the magnetic field h whenever the model has the
exponential weak mixing property. We also prove the exponential weak mixing
property whenever h=0. Together with known results on the regime
h=0,β<βc, this implies both analyticity and weak mixing in all the
domain of (β,h) outside of the transition line [βc,∞)×{0}. The proof of analyticity uses a graphical representation of the Glauber
dynamic due to Schonmann and cluster expansion. The proof of weak mixing uses
the random cluster representation.Comment: Updated Bibliograph