Bethe lattice

A Bethe lattice with coordination number z = 3

A Bethe lattice, introduced into the physics literature by Hans Bethe in 1935, is an infinite connected cycle-free graph where the vertices all have the same valence. In the mathematical literature a Bethe lattice is called a regular tree. In such a graph, each node is connected to z neighbours; in the physics literature, z is called the coordination number. In the mathematical literature z is called the degree of the regular tree. With one node chosen as root, all other nodes are seen to be arranged in shells around this root node, which is then also called the origin of the lattice. The number of nodes in the kth shell is given by

${\displaystyle \,N_{k}=z(z-1)^{k-1}{\text{ for }}k>0.}$

(Note that the Bethe lattice is actually an unrooted tree, since any vertex will serve equally well as a root.)

In some situations the definition is modified to specify that the root node has z − 1 neighbors.^{[citation needed]}

Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often exactly solvable. The solutions are related to the often used Bethe approximation for these systems.

Relation to Cayley graphs and Cayley trees[]

A Bethe graph of even coordination number 2n is isomorphic to the unoriented Cayley graph of a free group of rank n with respect to a free generating set.

Lattices in Lie groups[]

Bethe lattices also occur as the discrete subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups. As such, they are also lattices in the sense of a lattice in a Lie group.

See also[]

• Crystal

References[]

• Bethe, H. A. (1935). "Statistical theory of superlattices". Proc. Roy. Soc. Lond. A. 150 (871): 552–575. Bibcode:1935RSPSA.150..552B. doi:10.1098/rspa.1935.0122. Zbl 0012.04501.
• Baxter, Rodney J. (1982). Exactly solved models in statistical mechanics. Academic Press. ISBN 0-12-083182-1. Zbl 0538.60093.
• Ostilli, M. (2012). "Cayley Trees and Bethe Lattices, a concise analysis for mathematicians and physicists". Physica A. 391 (12): 3417. arXiv:1109.6725. Bibcode:2012PhyA..391.3417O. doi:10.1016/j.physa.2012.01.038. S2CID 119693543.