Weak Mixing and Analyticity of the Pressure in the Ising Model

Abstract

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 $h\ne 0$. 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.

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