Renormalization group transformations near the critical point: Some rigorous results

We consider renormalization group (RG) transformations for classical Ising-type lattice spin systems in the infinite-volume limit. Formally, the RG maps a Hamiltonian H into a renormalized Hamiltonian H', exp(-H(')(sigma('))) = Sigma(sigma)T(sigma,sigma('))exp(-H(sigma)), where T(sigma, sigma') denotes a specific RG probability kernel, Sigma(')(sigma)T(sigma,sigma(')) = 1, for every configuration sigma. With the help of the Dobrushin uniqueness condition and standard results on the polymer expansion, Haller and Kennedy gave a sufficient condition for the existence of the renormalized Hamiltonian in a neighborhood of the critical point. By a more complicated but reasonably straightforward application of the cluster expansion machinery, the present investigation shows that their condition would further imply a band structure on the matrix of partial derivatives of the renormalized interaction with respect to the original interaction. This in turn gives an upper bound for the RG linearization. (C) 2011 American Institute of Physics. [doi:10.1063/1.3660381]