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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 Zd\mathbb{Z}^d is analytic as a function of both the inverse temperature β\beta and the magnetic field hh whenever the model has the exponential weak mixing property. We also prove the exponential weak mixing property whenever h0h\neq 0. Together with known results on the regime h=0,β<βch=0,\beta<\beta_c, this implies both analyticity and weak mixing in all the domain of (β,h)(\beta,h) outside of the transition line [βc,)×{0}[\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.Comment: Updated Bibliograph

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