Quarkyonic Percolation and deconfinement at finite density and number of colors

We examine the interplay between the percolation and the deconfinement phase transitions of Yang-Mills matter at finite temperature, quark chemical potential $\mu_Q$ and number of colors $N_c$. We find that, whereas the critical $N_c$ for percolation goes down with density, the critical $N_c$ for confinement generally goes up. Because of this, Yang-Mills matter falls into two qualitatively different regimes: the "low-$N_c$ limit", where percolation does not occur because matter deconfines before it percolates, and the high-$N_c$ limit, where there are three distinct phases characterizing Yang-Mills matter at finite temperature and density: confined, deconfined and confined but percolating matter. The latter can be thought of as the recently conjectured "quarkyonic phase". We attempt to estimate the critical $N_c$, to see if the percolating phase can occur in our world. We find that, while percolation will not occur at normal nuclear density as in the large-$N_c$ limit, a sliver of the phase diagram in $N_c$, energy density and baryonic density where percolation occurs while confinement persists is possible. We conclude by speculating on the phenomenological properties of such percolating "quarkyonic" matter, suggest avenues to study it quantitatively, and look for it in experiment.