Bethe lattice

Remarks

Fix a vertex $o$, the root. For $n \ge 1$ there are exactly $q (q - 1)^{n - 1}$ vertices at distance $n$ from $o$, so the number of vertices grows exponentially in the distance when $q \ge 3$. As a result, for $q \ge 3$, in a finite ball the boundary vertices make up a fixed positive fraction of all vertices. So statistical models on it are studied deep inside the tree, away from the boundary. Since the tree has no cycles, models such as the Ising model can be solved exactly on it by recursion from the leaves. For the ferromagnetic Ising model with coupling $J > 0$ and no field, the critical inverse temperature $\beta_c$ satisfies $\tanh(\beta_c J) = 1/(q - 1)$. This solution coincides with the Bethe–Peierls approximation for a lattice with coordination number $q$, which is the origin of the name. For $q \ge 3$ it is the Cayley graph of a free product of $q$ copies of $\Z/2\Z$.