We consider
1/d expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate
1/d expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in
1/d. The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in
1/d.We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum -- and, in particular, classical -- interactions. Our approach is based on the use of ``decoupling parameters", advocated by Park [34], which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a
β-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small
β. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams [2]. As an application we show that our lower bound of the radius of
β-analyticity is larger than Park's for quantum systems two-body interactions.