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

QCD in the heavy dense regime: Large $N_c$ and quarkyonic matter