Bethe lattice

A Bethe lattice, introduced by Hans Bethe in 1935, is an infinite connected cycle-free graph where the vertices all have the same valence. I.e., each node is connected to z neighbours; z is called the coordination number. 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

A Bethe lattice with coordination number z = 3
${\displaystyle \,N_{k}=z(z-1)^{k-1}{\text{ for }}k>0.}$

(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

The Bethe lattice where each node is joined to 2n others is essentially the Cayley graph of a free group on n generators. It is an infinite Cayley tree.

A presentation of a group G by n generators corresponds to a surjective map from the free group on n generators to the group G, and at the level of Cayley graphs to a map from the Bethe lattice (with distinguished root corresponding to the identity) to the Cayley graph. This can also be interpreted (in algebraic topology) as the universal cover of the Cayley graph, which is not in general simply connected.

A Bethe lattice is defined by its coordination number. It is an unrooted tree, since every vertex is identical, with z neighbors. It also has no surface since it extends to infinity. On the other hand a Cayley tree has a root and a highly non-negligible surface.

The root of a Cayley tree, like all its nodes except the leaves, has valence z (the leaves have valence 1). An infinite Cayley tree has no leaves, so all its nodes have valence z. Define the connectivity of a node as the number of edges connected to it. Since there are no self-edges, and at most one edge connecting any two nodes, this is the same as the number of distinct nodes to which it is connected by an edge. Thus for a (finite) Cayley tree the average connectivity c of a node is the same as its average degree, viz.

${\displaystyle \,c={\frac {2E}{V}}={\frac {2(V-1)}{V}}=2-{\frac {2}{V}};}$

whereas the average connectivity of a Bethe lattice (infinite Cayley tree) is just z.

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.