  We consider a binary mixture of colloid and polymer particles with positions
on a simple cubic lattice. Colloids exclude both colloids and polymers from
nearest neighbor sites. Polymers are treated as effective particles that are
mutually non-interacting, but exclude colloids from neighboring sites; this is
a discrete version of the (continuum) Asakura-Oosawa-Vrij model. Two
alternative density functionals are proposed and compared in detail. The first
is based on multi-occupancy in the zero-dimensional limit of the bare model,
analogous to the corresponding continuum theory that reproduces the bulk fluid
free energy of free volume theory. The second is based on mapping the polymers
onto a multicomponent mixture of polymer clusters that are shown to behave as
hard cores; the corresponding (Highlander) property of the extended model in
strong confinement permits direct treatment with lattice fundamental measure
theory. Both theories predict the same topology for the phase diagram with a
continuous fluid-fcc freezing transition at low polymer fugacity and, upon
crossing a tricritical point, a first-order freezing transition for high
polymer fugacities with rapidly broadening density jump.
