  We investigate the properties of the half-filling point in lattice QCD
(LQCD), in particular the disappearance of the sign problem and the emergence
of an apparent particle-hole symmetry, and try to understand where these
properties come from by studying the heavy-dense fermion determinant and the
corresponding strong-coupling partition function (which can be integrated
analytically). We then add in a first step an effective Polyakov loop gauge
action (which reproduces the leading terms in the character expansion of the
Wilson gauge action) to the heavy-dense partition function and try to analyze
how some of the properties of the half-filling point change when leaving the
strong coupling limit. In a second step, we take also the leading
nearest-neighbor fermion hopping terms into account (including gauge
interactions in the fundamental representation) and mention how the method
could be improved further to incorporate the full set of nearest-neighbor
fermion hoppings. Using our mean-field method, we also obtain an approximate
($\mu$,T) phase diagram for heavy-dense LQCD at finite inverse gauge coupling
$\beta$. Finally, we propose a simple criterion to identify the chemical
potential beyond which lattice artifacts become dominant.
