Why Loops Don't Matter

TL;DR

This paper demonstrates that for certain ferromagnetic spin models, the presence of loops in random graphs does not affect their thermodynamic behavior, as their saddle point equations match Bethe lattice recursion fixed points.

Contribution

It reveals a direct correspondence between saddle point equations in random graph models and recursion relations in Bethe lattice models, showing loops are irrelevant in the thermodynamic limit.

Findings

Ratios of saddle point equations match Bethe lattice fixed points

Loops in random graphs do not influence thermodynamic properties

Correspondence extends to multi-spin interaction models on cacti

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.