Expansions in the inverse of the coordination number z have proved very useful in condensed matter physics. In this paper, we review the formalism of the 1/z expansion for magnetic systems and arrays of Josephson junctions. While many variations of the 1/z expansion have been developed, we concentrate on unrenormalized expansions, where the expansion coefficients are independent of the coordination number but may depend on dimension. The 1/z expansion has been used to study both first- and second-order phase transitions in a variety of systems, including the Ising and Heisenberg models, the Blume-Capel model, and Josephson-junction arrays. This expansion has also been used to study the dynamics of a Heisenberg ferromagnet. Although they sacrifice the rigor of the 1/z expansion, resummations or rearrangements of the 1/z expansion have proved useful to study critical behavior.
Lack of phase transitions in staggered magnetic systems. A comparison of uniqueness criteria
