Solvability of the system of integral equations of lattice models of statistical mechanics

The paper is a review of results on the solvability of a system of integral equations, which is an analog of the Kirkwood–Salzburg equations for an infinite set of partial probability distributions of Gibbs random sets on $\mathbb{Z}^d$ corresponding to lattice gas models of equilibrium statistical mechanics with a pair interaction potential $U$. We study the relationship between the solvability of the system and the location of zeros of the partition functions $Q_\Lambda(z)$ of models.