Global Aspects of the Renormalization Group Flows of Dyson's Hierarchical Model

Dyson's hierarchical model (HM) is a lattice scalar model for which the effective potential can be calculated very accurately using the renormalization group method. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigorous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the HM and exact renormalization group equations (ERGE) in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large-N limit and the complex singularities of the critical potential calculable in this limit. The improvement of the hierarchical approximation is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review article is to present a configuration space counterpart, suitable for lattice formulations, of ERGE formulated in momentum space