Conditional Ising equilibrium phase predictions

Assume Countable Choice and adopt Interacting classical Gibbs equilibrium adoption for the specified classical Ising model. In the SI interpretation of Classical Gibbs quantities and SI reference conventions, $\beta =(k_{B}T{)}^{-1}$ is in ${\mathrm{J}}^{-1}$, $J$ is the coupling energy $J_{\mathrm{I}}$ in joules, and $h$ is an energy bias per dimensionless spin in joules, not a magnetic-field strength; hence $\beta J$ and $\beta h$ are dimensionless. The conditional model predictions are: (i) for a one-dimensional nearest-neighbor chain with finite $\beta >0$, $J,h\in \mathbb{R}$, the DLR equilibrium state is unique; (ii) for a translation-invariant finite-range Ising pair interaction with symmetric zero-diagonal couplings $J_{ij}=J_{ji}$, $J_{ii}=0$ and constant finite field, the sufficient condition ${\sup }_i{\sum }_j\tanh (\beta \mathrm{\mid }{J}_{ij}\mathrm{\mid })<1$ gives a unique DLR equilibrium state; and (iii) for the two-dimensional nearest-neighbor ferromagnet at $h=0$ and $\beta J\ge 2$, there are distinct stationary DLR equilibrium states with opposite nonzero magnetization and the pressure is nondifferentiable in $h$ at zero. These deductions are restricted to the stated classical model; they do not report a measured transition or infer a phase diagram between the sufficient regimes.