methods[@kadanoff77; @ginsparg88; @boyanovsky89a; @lecheminant02] combined with advanced classical Monte Carlo (MC) simulations[@krauth03; @liu04] to probe the correlations in the [*doped*]{} RK wave functions, investigate the phase diagram and its phase transitions. The methods used here can be readily generalized to the case of non-bipartite lattices, for which a number of important results have been published.[@moessner01a; @fendley02; @trousselet07] In Section \[sec:hamiltonians\], we describe the construction of two generalized quantum dimer RK Hamiltonians that we used in our study. A similar but independent construction has been presented by Castelnovo and coworkers[@Castelnovo06] and by Poilblanc and coworkers[@Poilblanc06]. The RK wave functions of these generalizations of the quantum dimer model have either a fixed number of holes or a variable number of holes and a fixed hole fugacity. The ground-state wave functions of both models at their associated RK points correspond to a canonical dimer-monomer system in the canonical and grand-canonical ensembles respectively. Near the end of the paper, in Section \[sec:hole-int\], we introduce a third Hamiltonian, with an associated RK wave function, to study the effects of hole interactions which compete with phase separation at the first-order transitions that we find for both models.

In Sections \[sec:mean-field\] and \[sec:exact-critical-behavior\] we study the correlations 