Equivalence of ferromagnetic spin models on trees and random graphs

Recently we have found identical behaviour for various spin models on 'thin' random graphs - Feynman diagrams - and the corresponding Bethe lattices. In this paper we observe that the ratios of the saddle-point equations in the random graph approach are identical to the fixed point(s) of the recursion relations which are used to solve the models on the Bethe lattice. The loops in the random graphs thus have no influence on the thermodynamic limit for such ferromagnetic spin models. We consider the correspondence explicitly for Ising and q state Potts models and also note that multispin interaction models on cacti admit a similar correspondence with a randomized version of the cacti graphs which contain loops.