  A family of models is proposed to describe the motion of holes in a
fluctuating quantum dimer background on the square lattice. Following
Castelnovo et al. [Ann. Phys. (NY) 318, 316 (2005)], a generalized
Rokhsar-Kivelson Hamiltonian at **finite doping** which can be mapped on a
**doped** interacting classical dimer model is constructed. A simple physical
extension of this model is also considered. Using numerical computations and
simple considerations based on the above exact mapping, we determine the phase
diagram of the model showing a number of quantum phases typical of a doped Mott
insulator. The two-hole correlation function generically exhibits short-range
or long-range algebraic correlations in the solid (columnar) and liquid
(critical) phases of the model, respectively. Evidence for an extended region
of a doped VBS phase exhibiting holon pairing but **no** phase separation is
given. In contrast, we show that hole deconfinement occurs in the staggered
dimer phase.
