Gas of “connected configurations” and allowance for the “hard-core” potential of contours in the Mayer expansion of a gas of lattice-model contours

A theorem is proved that makes it possible to take into account the “hard-core” potential of contours and reduce the study of the convergence of the Mayer expansions of the gas of contours to the remaining part of the interaction. In particular, for a model with nearest-neighbor interaction, in which
$$ U(\alpha)=\sum_{|x-y|=1}\varepsilon(\alpha(x)\alpha^{-1}(y)), $$
$\alpha(x)$ takes values in the discrete group $G$ with identity $e$, $\varepsilon(\alpha)=\varepsilon(\alpha^{-1})$ $\forall\alpha\ne e$, $\varepsilon(e)=0$ and
$$ \sum_{\alpha\in G\setminus e}\exp\{-\beta U(\alpha)\} \underset{\beta\to\infty}\longrightarrow0, $$
the existence is proved of not less than $|G|$ ($|G|\leqslant\infty$) limit Gibbs distributions, which are small perturbations of the ground states $\alpha(x)=\alpha_0\in G$.