  QCD is expected to have a rich phase structure. It is empirically known to be
difficult to access low temperature and nonzero chemical potential $\mu$
regions in lattice QCD simulations. We address this issue in a lattice QCD with
the use of a dimensional reduction formula of the fermion determinant.
  We investigate spectral properties of a reduced matrix of the reduction
formula. Lattice simulations with different lattice sizes show that the
eigenvalues of the reduced matrix follow a scaling law for the temporal size
$N_t$. The properties of the fermion determinant are examined using the
reduction formula. We find that as a consequence of the $N_t$ scaling law, the
fermion determinant becomes insensitive to $\mu$ as $T$ decreases, and
$\mu$-independent at T=0 for $\mu<m_\pi/2$.
  The $N_t$ scaling law provides two types of the low temperature limit of the
fermion determinant: (i) for low density and (ii) for high-density. The fermion
determinant becomes real and the theory is free from the sign problem in both
cases. In case of (ii), QCD approaches to a theory, where quarks interact only
in spatial directions, and gluons interact via the ordinary Yang-Mills action.
The partition function becomes exactly $Z_3$ invariant even in the presence of
dynamical quarks because of the absence of the temporal interaction of quarks.
  The reduction formula is also applied to the canonical formalism and Lee-Yang
zero theorem. We find characteristic temperature dependences of the canonical
distribution and of Lee-Yang zero trajectory. Using an assumption on the
canonical partition function, we discuss physical meaning of those temperature
dependences and show that the change of the canonical distribution and Lee-Yang
zero trajectory are related to the existence/absence of $\mu$-induced phase
transitions.
