Cayley Trees and Bethe Lattices, a concise analysis for mathematicians   and physicists
  We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important - and different - ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community. 
Massimo Ostilli
cond-mat.stat-mech cond-mat.dis-nn math-ph math.MP