  After combined character and hopping expansions and integration over the
spatial gauge links, lattice QCD reduces to a three-dimensional $SU(3)$
Polyakov loop model with complicated interactions. A simple truncation of the
effective theory is valid for heavy quarks on reasonably fine lattices and can
be solved by linked cluster expansion in its effective couplings. This was used
ealier to demonstrate the onset transition to baryon matter in the cold and
dense regime. Repeating these studies for general $N_c$, one finds that for
large $N_c$ the onset transition becomes first-order, and the pressure scales
as $p\sim N_c$ through three consecutive orders in the hoppoing expansion.
These features are consistent with the formal definition of quarkyonic matter
given in the literature. We discuss the implications for $N_c=3$ and physical
QCD.
