Why Loops Don't Matter

Abstract

In a series of papers we have found identical behaviour for various spin models on thin random graphs - Feynman diagrams - and the corresponding Bethe lattices. In this note we observe that in all cases the ratios of various saddle point equations in the random graph approach are identical in form 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 in the thermodynamic limit for such ferromagnetic spin models. We consider the correspondence explicitly for Ising and q state Potts models and also note that multi spin interaction models on cacti admit a similar correspondence with a randomised version of the cacti graphs which contain loops.