Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator Approach

In this paper we use a recent version of the Ruelle-Perron-Frobenius Theorem to compute, in terms of the maximal eigendata of the Ruelle operator, the pressure derivative of translation invariant spin systems taking values on a general compact metric space. On this setting the absence of metastable states for continuous potentials on one-dimensional one-sided lattice is proved. We apply our results, to show that the pressure of an essentially one-dimensional Heisenberg-type model, on the lattice $\mathbb{N}\times \mathbb{Z}$, is Fr\'echet differentiable, on a suitable Banach space. Additionally, exponential decay of the two-point function, for this model, is obtained for any positive temperature.