  The cold and dense regime of the QCD phase diagram is to this day
inaccessible to first principle lattice calculations owing to the sign problem.
Here we present progress of an ongoing effort to probe this particularly
difficult regime utilising a dimensionally reduced effective lattice theory
with a significantly reduced sign problem. The effective theory is derived by
combined character and hopping expansion and is valid for heavy quarks near the
continuum. We show an extension of the effective theory to order $u^5\kappa^8$
in the cold regime. A linked cluster expansion is applied to the effective
theory resulting in a consistent mechanism for handling the effective theory
fully analytically. The new results are consistent with the ones from
simulations confirming the viability of analytic methods. Finally we resum the
analytical result which doubles the convergence region of the expansion.
