In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite symmetric regular tree where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature by Hans Bethe in 1935. In such a graph, each node is connected to z neighbors; the number z is called either the coordination number or the degree, depending on the field.

Due to its distinctive topological structure, the statistical mechanics of lattice models on this graph are often easier to solve than on other lattices. The solutions are related to the often used Bethe ansatz for these systems.

Basic properties

When working with the Bethe lattice, it is often convenient to mark a given vertex as the root, to be used as a reference point when considering local properties of the graph.

Sizes of layers

vertices at a distance 1.

In statistical mechanics

The Bethe lattice is of interest in statistical mechanics mainly because lattice models on the Bethe lattice are often easier to solve than on other lattices, such as the two-dimensional square lattice. This is because the lack of cycles removes some of the more complicated interactions. While the Bethe lattice does not as closely approximate the interactions in physical materials as other lattices, it can still provide useful insight.

Exact solutions to the Ising model

The Ising model is a mathematical model of ferromagnetism representing an external magnetic field.

The Ising model on the Bethe lattice is defined by the partition function

Magnetization

In order to compute the local magnetization, we can break the lattice up into several identical parts by removing a vertex. This gives us a recurrence relation which allows us to compute the magnetization of a Cayley tree with n shells (the finite analog to the Bethe lattice) as

satisfy the recurrence relation

case when the system is ferromagnetic, the above sequence converges, so we may take the limit to evaluate the magnetization on the Bethe lattice. We get

1. where x.

.

Free energy

The free energy f at each site of the lattice in the Ising Model is given by

1. ,

is as before.

In mathematics

Return probability of a random walk

The probability that a random walk away. We have the recurrence relation

, we get

1. .

, which is the value we want.

Note that this in stark contrast to the case of random walks on the two-dimensional square lattice, which famously has a return probability of 1. Such a lattice is 4-regular, but the 4-regular Bethe lattice has a return probability of 1/3.

Number of closed walks

th Catalan number.

choices for an outward step from the starting vertex, which happens at the beginning and any number of times during the walk. The exact number of walks is trickier to compute, and is given by the formula

is the Gauss hypergeometric function.

be its adjacency matrix be the second largest absolute value of an eigenvalue, we have

This is a rather interesting result in the study of (n,d,λ)-graphs.

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.

Hyperbolic geometry

apeirogonal tiling of the hyperbolic plane.

See also

• Crystal

References