Bethe lattice

A Bethe lattice or Cayley tree (though the two are not completely equivalent, see below), introduced by Hans Bethe in 1935, is a connected cycle-free graph where each node is connected to z neighbours, where z is called the coordination number. It can be seen as a tree-like structure emanating from a central node, with all the nodes arranged in shells around the central one. The central node may be called the root or origin of the lattice. The number of nodes in the kth shell is given by

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

In some situations the definition is modified to specify that the root node has z − 1 neighbours.

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.

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Relation to Cayley graphs

Further information: Cayley graph

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

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 Cayley tree 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.

The distinction between a Bethe lattice and a Cayley tree is that the former is the thermodynamic limit of the latter. Hence in Cayley trees, surface effects become important.

Lattices in Lie groups

Bethe lattices also occur as the discrete group 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.

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