Renormalization Group in the Uniqueness Region: Weak Gibbsianity and Convergence

We analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature. We prove weak Gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense. Since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system: the correlation length or the critical length for metastability, or both. Thus, to perturbatively treat the problem we have to use a scale–adapted expansion. Moreover, such a model below the critical temperature resembles a disordered system in the presence of Griffiths' singularity. Then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached. The authors acknowledge the support of Cofinanziamento MIUR.