The large field renormalization operation for classical \(N\)-vector models (Q1282100)

Classical \(N\)-vector models have been the subject of intensive studies in the past. Balaban continues his analysis of the renormalization group \((ST\)-transformation to be precise) applied to densities. For previous work and results with regard to this problem, see [ibid. 167, 103--154 (1995; Zbl 0811.60100), ibid. 182, 675--721 (1996; Zbl 0869.60095), and ibid. 198, 1--45 (1998; Zbl 0932.81020)]. Related work of the author on the \(N\)-vector model may be found in Lect. Notes Phys. 446, 201--218 (1995; Zbl 0825.81011) and Commun. Math. Phys. 175, 607--642 (1996; Zbl 0838.60092). The precise description of renormalization group flow for large-field densities leads to quite elaborate constructions and estimates. These estimates form the main bulk of the present work. The goal is to prove that the \(ST\)-transformation, when applied to a density, preserves a set of ``inductive'' hypotheses. These hypotheses give sufficient control over integrals needed for the construction of the generating functional for the correlations functions to all orders.