Uniqueness of the limit Gibbs distribution in one-dimensional classical systems

Uniqueness of the limit Gibbs distribution is proved for the one-dimensional latticesystems, in which the slow decreasing of the inter-particle interaction is allowed. The main restriction on the interaction potential $U(c)$ is
$$ \sum_{c\colon0\in c,\,\operatorname{diam}\{c\}=K}\operatorname{diam}\{c\}|U(c)|<B\ln\ln K, $$
where $c=\{x_1,\dots,x_n\}$ is an arbitrary configuration of particles on the lattice and $B$ is some sufficiently small constant.