The hierarchical model and the renormalization group

P. Coullet, E. Tirapegui

J. Goettker-Schnetmann

Generalizing methods developed by Pinn, Pordt and Wieczerkowski for the hierarchical model with one component (N=1) and dimensions d between 2 and 4 we compute O(N)-symmetric fixed points of the hierarchical renormalization group equation for some N and d with 0 < d < 4 and -2 <= N <= 20. The spectra of the linearized RG equation at the fixed points are calculated and the critical exponents νare extracted from the spectrum and compared to Borel-Pade-resummed ε-expansion.

Aurélien Decelle, Giorgio Parisi A, Jacopo Rocchi

The renormalization group (RG) methods are still far from being completely understood in quenched disordered systems. In order to gain insight into the nature of the phase transition of these systems, it is common to investigate simple models. In this work we study a real-space RG transformation on the Dyson hierarchical lattice with a random field, which leads to a reconstruction of the RG flow and to an evaluation of the critical exponents of the model at T=0. We show that this method gives very accurate estimations of the critical exponents by comparing our results with those obtained by some of us using an independent method.

S. Niermann

We consider the problem of calculating the nonuniversal parameters entering into the scaling laws of renormalized quantities. We examine the renormalization group transformation of Dyson's hierarchical model and a simplified version of the model. We introduce the hierarchical model and explain the use of approximations which aid in extracting numerical information about the susceptibility. We then use these approximations to relate the hierarchical model to a simplified, one- parameter model. In the simplified model, we can express the magnetic susceptibility in terms of dual expansions, one about the high-temperature fixed point, and the other about the critical point. With the introduction of two dual scaling variables which transform covariantly under the renormalization group transformation, we are able to calculate all of the nonuniversal quantities. Extensions of these techniques to multi-parameter models, such as the hierarchical model, are discussed.