Phase diagrams and singularity at the point of a phase transition of the first kind in lattice gas models

Contour representation of the statistical sum of a $d$-dimensional $(d\geq 2)$ lattice gas models with the pair finite potential is studied. On the complex plane of the chemical potential $\mu$ the equation is found for the line of the phase transition of the first kind between two phases with the arbitrarily complicated periodic ordering of atoms of the lattice in the ground state in the assumption of the validity of Peierls' hypothesis. Inverse temperature $\beta=T^{-1}$ is also considered as complex and having a sufficiently large real part. Analyticity of the pressure outside the transition line is proved as well as the existence of finite limits of all derivatives of the pressure at $\mu$ going to the transition line. The explicit form of asymptotics of the limiting values of the derivatives $d^kp/d\mu^k\sim(k!)^{d/(d-1)}$ for real $\beta, \mu$ and large $k$ is found. It proves that the point of the first kind phase transition is an essentially singular point. Relation of the resut obtained to the properties of the system in metastable state is discussed.