  The linked cluster expansion has been shown to be highly efficient in
calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D
classical and quantum lattice models. In this article, we extend the linked
cluster method to the Cayley tree and its boundaryless cousin the Bethe
lattice. We aim to (a) develop the linked cluster expansion for these lattices,
a novel application, and (b) to further understand the surprising convergence
efficiency of the linked cluster method, as well as its limitations. We obtain
several key results. First, we show that for nearest-neighbor Hamiltonians of a
specific form, all finite treelike clusters can be mapped to one dimensional
finite chains. We then show that the qualitative distinction between the Cayley
tree and Bethe lattice appears due to differing lattice constants that is a
result of the Bethe lattice being boundaryless. We use these results to obtain
the explicit closed-form formula for the zero-field susceptibility for the
entire disordered phase up to the critical point for Bethe lattices of
arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also
obtain the exact zero field partition function for the Ising model on both
trees with only the two smallest clusters, similar to the 1D chain. Finally,
these results achieve a direct comparison between an infinite lattice with a
nonnegligible boundary and one without any boundary, allowing us to show that
the linked cluster expansion eliminates boundary terms at each order of the
expansion, answering the question about its surprising convergence efficiency.
We conclude with some ramifications of these results, and possible
generalizations and applications.
