The Cayley Tree

Abstract

In the previous chapter, we saw that the MF approximation becomes correct in the limit of infinite spatial dimensionality. In this limit, fluctuations in the field sensed by one spin (in the Ising model) becomes small and our neglect of correlated fluctuations seems appropriate. In this chapter, we consider the construction of exact solutions to statistical problems on the Cayley tree. As we will see in a moment, the infinite Cayley tree (which is sometimes called a "Bethe lattice") provides a realization of infinite spatial dimensionality. The Cayley tree is a lattice in the form of a tree (i.e., it has no loops) which is recursively constructed as follows. One designates a central or seed site as the zeroth generation of the lattice. The first generation of the lattice consists of z sites which are neighboring to the seed site. Each first-generation site also has z nearest neighbors: one already present in the zeroth generation and σ ≡ z − 1 new sites added in the second generation of the lattice. The third generation of sites consists of the σ new sites neighboring to the second-generation sites. There are thus z first-generation sites and zσ (k−1) kth generation sites. In Fig. 14.1 we show four generations of a Cayley tree with z = 3. Note that a tree with z = 2 is just a linear chain. We have previously remarked that mean-field theory should become progressively more accurate as the coordination number increases. As we will see below, a more precise statement is that for systems with short-range interactions, the critical exponents become equal to their mean-field values at high spatial dimensionality. It is therefore of interest to determine the spatial dimensionality of the Cayley tree. There are at least two ways to address this question. One way is to study how the number N (r) of lattice points in a spherical volume of radius r increases for large r. If we consider a tree with k generations the number of sites is N (k) = 1 + z + zσ + zσ 2 +. .. zσ k−1 = 1 + z σ k − 1 σ − 1. (14.1)