Absence of renormalization group pathologies near the critical temperature. Two examples

We consider real-space renormalization group transformations for Ising-type systems which are formally defined by exp[ - H′(σ′)] = ∑ T(σ, σ′) exp[ - H(σ)] where T(σ, σ′) is a probability kernel, i.e., ∑_σ′, T(σ, σ′) = 1, for every configuration σ. For each choice of the block spin configuration σ′, let μ_σ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional to T(σ, σ′) exp[ - H(σ)]. We give a condition which is sufficient to imply that the renormalized Hamiltonian H′ is defined. Roughly speaking, the condition is that the collection of measures μ_σ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacing b = 2 on the square lattice with β < 1.36β_c and the Kadanoff transformation with parameter p on the triangular lattice in a subset of the β, p plane that includes values of β greater than β_c.