Weak Mixing and Analyticity of the Pressure in the Ising Model

We prove that the pressure (or free energy) of the finite range ferromagnetic Ising model on $\mathbb{Z}^d$ is analytic as a function of both the inverse temperature $\beta$ and the magnetic field $h$ whenever the model has the exponential weak mixing property. We also prove the exponential weak mixing property whenever $h\neq 0$. Together with known results on the regime $h=0,\beta<\beta_c$, this implies both analyticity and weak mixing in all the domain of $(\beta,h)$ outside of the transition line $[\beta_c,\infty)\times \{0\}$. 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.