A note on cactus trees: Variational vs. recursive approach

In this article we deal with the variational approach to cactus trees (Husimi trees) and the more common recursive approach, that are in principle equivalent for finite systems. We discuss in detail the conditions under which the two methods are equivalent also in the analysis of infinite (self-similar) cactus trees, usually investigated to the purpose of approximating ordinary lattice systems. Such issue is hardly ever considered in the literature. We show (on significant test models) that the phase diagram and the thermodynamic quantities computed by the variational method, when they deviates from the exact bulk properties of the cactus system, generally provide a better approximation to the behavior of a corresponding ordinary system. Generalizing a property proved by Kikuchi, we also show that the numerical algorithm usually employed to perform the free energy minimization in the variational approach is always convergent.