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In this paper, we study perturbations of the $d$-dimensional Ising model for$d\geq 2$, including long range ones to which the Pirogov-Sinai theory is notapplicable. We show that the uniqueness of the equilibrium state of the Isingmodel at high temperature and the coexistence of equilibrium states at lowtemperature are preserved by spin-flip symmetric perturbations.

Stability of uniqueness and coexistence of equilibrium states of the Ising model under long range perturbations