Renormalization-Group Transformations Under Strong Mixing Conditions: Gibbsianness and Convergence of Renormalized Interactions

In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite-range lattice gases, when suitable strong mixing conditions are satisfied. Using a block decimation procedure, cluster expansion, and detailed comparison between statistical ensembles, we are able to prove Gibbsianness and convergence to a trivial (i.e., Gaussian and product) fixed point. Our results apply to the 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.