Bethe lattice
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.
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.
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.
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.
The Ising model is a mathematical model of ferromagnetism, in which the magnetic properties of a material are represented by a "spin" at each node in the lattice, which is either +1 or -1.
representing the strength of the interaction between adjacent nodes, and a constant
The Ising model on the Bethe lattice is defined by the partition function
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
case when the system is ferromagnetic, the above sequence converges, so we may take the limit to evaluate the magnetization on the Bethe lattice.
The free energy f at each site of the lattice in the Ising Model is given by
[1] The probability that a random walk on a Bethe lattice of degree
be the probability of returning to our starting point if we are a distance
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.
One can easily bound the number of closed walks of length
starting at a given vertex of the Bethe Lattice with degree
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
times the number of closed walks on the Bethe lattice with degree
that start at a given vertex and only go back on paths that were already tread.
, as we can make use of cycles to create additional walks.
be the second largest absolute value of an eigenvalue, we have
for which the second largest absolute value of an eigenvalue is at most
This is a rather interesting result in the study of (n,d,λ)-graphs.
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.
Bethe lattices also occur as the discrete subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups.
apeirogonal tiling of the hyperbolic plane form a Bethe lattice of degree
