Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists

Abstract

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.

Heisenberg antiferromagnet on the Husimi lattice

“…The Husimi tree therefore has very strong finite-size effects and the physical properties of any model defined on the finite tree may be very different from those of a model on the infinite lattice. Normally a model defined on the Husimi lattice is much more suitable to represent a real physical system than a model on the Husimi tree [39].…”

Mapping the q-voter model: From a single chain to complex networks

“…For the sake of simplicity let us assume that our network has the topology of a Bethe lattice [40] with the coordination number k. For convenience, we took a modified definition of the Bethe lattice with the central node having only k − 1 neighbors (Fig. 7).…”

Phase diagram of the isotropic spin-32model on thez=3Bethe lattice

“…This lattice is infinite by definition and thus there are no surface effects. 27 Note that the thermodynamic limit of the Cayley tree and the Bethe lattice are not equivalent. This inequivalence is rooted in the large number of surface sites contained in any Cayley tree.…”