Bethe lattice

Once a vertex is marked as the root, we can group the other vertices into layers based on their distance from the root. The number of vertices at a distance ${\displaystyle d>0}$ from the root is ${\displaystyle z(z-1)^{d-1}}$, as each vertex other than the root is adjacent to ${\displaystyle z-1}$ vertices at a distance one greater from the root, and the root is adjacent to ${\displaystyle z}$ vertices at a distance 1.

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. The model is also equipped with a constant ${\displaystyle K}$ representing the strength of the interaction between adjacent nodes, and a constant ${\displaystyle h}$ representing an external magnetic field.

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

${\displaystyle Z=\sum _{\{\sigma \}}\exp \left(K\sum _{(i,j)}\sigma _{i}\sigma _{j}+h\sum _{i}\sigma _{i}\right).}$

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

${\displaystyle M={\frac {e^{h}-e^{-h}x_{n}^{q}}{e^{h}+e^{-h}x_{n}^{q}}},}$

where ${\displaystyle x_{0}=1}$ and the values of ${\displaystyle x_{i}}$ satisfy the recurrence relation

${\displaystyle x_{n}={\frac {e^{-K+h}+e^{K-h}x_{n-1}^{q-1}}{e^{K+h}+e^{-K-h}x_{n-1}^{q-1}}}}$

In the ${\displaystyle K>0}$ 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

${\displaystyle M={\frac {e^{2h}-x^{q}}{e^{2h}+x_{q}}},}$ where x is a solution to ${\displaystyle x={\frac {e^{-K+h}+e^{K-h}x^{q-1}}{e^{K+h}+e^{-K-h}x^{q-1}}}}$.

There are either 1 or 3 solutions to this equation. In the case where there are 3, the sequence ${\displaystyle x_{n}}$ will converge to the smallest when ${\displaystyle h>0}$ and the largest when ${\displaystyle h<0}$.

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

${\displaystyle {\frac {f}{kT}}={\frac {1}{2}}[-Kq-q\ln(1-z^{2})+\ln(z^{2}+1-z(x+1/x))+(q-2)\ln(x+1/x-2z)]}$,

where ${\displaystyle z=\exp(-2K)}$ and ${\displaystyle x}$ is as before.[1]

The probability that a random walk on a Bethe lattice of degree ${\displaystyle z}$ starting at a given vertex eventually returns to that vertex is given by ${\displaystyle {\frac {1}{z-1}}}$. To show this, let ${\displaystyle P(k)}$ be the probability of returning to our starting point if we are a distance ${\displaystyle k}$ away. We have the recurrence relation

${\displaystyle P(k)={\frac {1}{z}}P(k-1)+{\frac {z-1}{z}}P(k+1)}$

for all ${\displaystyle k>1}$, as at each location other than the starting vertex there are ${\displaystyle z-1}$ edges going away from the starting vertex and 1 edge going towards it. Summing this equation over all ${\displaystyle k>1}$, we get

${\displaystyle \sum _{k=1}^{\infty }P(k)={\frac {1}{z}}P(0)+{\frac {1}{z}}P(1)+\sum _{k=2}^{\infty }P(k)}$.

We have ${\displaystyle P(0)=1}$, as this indicates that we have just returned to the starting vertex, so ${\displaystyle P(1)=1/(z-1)}$, 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.[2] Such a lattice is 4-regular, but the 4-regular Bethe lattice has a return probability of 1/3.

One can easily bound the number of closed walks of length ${\displaystyle 2k}$ starting at a given vertex of the Bethe Lattice with degree ${\displaystyle z}$ from below. By considering each step as either an outward step (away from the starting vertex) or an inward step (toward the starting vertex), we see that any closed walk of length ${\displaystyle 2k}$ must have exactly ${\displaystyle k}$ outward steps and ${\displaystyle k}$ inward steps. We also may not have taken more inward steps than outward steps at any point, so the number of sequences of step directions (either inward or outward) is given by the ${\displaystyle k}$th Catalan number ${\displaystyle C_{k}}$. There are at least ${\displaystyle z-1}$ choices for each outward step, and always exactly 1 choice for each inward step, so the number of closed walks is at least ${\displaystyle (z-1)^{k}C_{k}}$.

This bound is not tight, as there are actually ${\displaystyle z}$ 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

${\displaystyle (z-1)^{k}C_{k}\cdot {\frac {z-1}{z}}\ _{2}F_{1}(k+1/2,1,k+2,4(z-1)/z^{2}),}$

where ${\displaystyle _{2}F_{1}(\alpha ,\beta ,\gamma ,z)}$ is the Gauss hypergeometric function.[3]

We may use this fact to bound the second largest eigenvalue of a ${\displaystyle d}$-regular graph. Let ${\displaystyle G}$ be a ${\displaystyle d}$-regular graph with ${\displaystyle n}$ vertices, and let ${\displaystyle A}$ be its adjacency matrix. Then ${\displaystyle {\text{tr }}A^{2k}}$ is the number of closed walks of length ${\displaystyle 2k}$. The number of closed walks on ${\displaystyle G}$ is at least ${\displaystyle n}$ times the number of closed walks on the Bethe lattice with degree ${\displaystyle d}$ starting at a particular vertex, as we can map the walks on the Bethe lattice to the walks on ${\displaystyle G}$ that start at a given vertex and only go back on paths that were already tread. There are often more walks on ${\displaystyle G}$, as we can make use of cycles to create additional walks. The largest eigenvalue of ${\displaystyle A}$ is ${\displaystyle d}$, and letting ${\displaystyle \lambda _{2}}$ be the second largest absolute value of an eigenvalue, we have

${\displaystyle n(d-1)^{k}C_{k}\leq {\text{tr}}A^{2k}\leq d^{2k}+(n-1)\lambda _{2}^{2k}.}$

This gives ${\displaystyle \lambda _{2}^{2k}\geq {\frac {1}{n-1}}(n(d-1)^{k}C_{k}-d^{2k})}$. Noting that ${\displaystyle C_{k}=(4-o(1))^{k}}$ as ${\displaystyle k}$ grows, we can let ${\displaystyle n}$ grow much faster than ${\displaystyle k}$ to see that there are only finitely many ${\displaystyle d}$-regular graphs ${\displaystyle G}$ for which the second largest absolute value of an eigenvalue is at most ${\displaystyle \lambda }$, for any ${\displaystyle \lambda <2{\sqrt {d-1}}.}$ 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. As such, they are also lattices in the sense of a lattice in a Lie group.