  We investigate the ground-state properties of trapped fermion systems
described by the Hubbard model with an external confining potential. We discuss
the universal behaviors of systems in different regimes: from few particles,
i.e. in dilute regime, to the trap thermodynamic limit.
  The asymptotic trap-size (TS) dependence in the dilute regime (increasing the
trap size l keeping the particle number N fixed) is described by a universal TS
scaling controlled by the dilute fixed point associated with the
metal-to-vacuum quantum transition. This scaling behavior is numerically
checked by DMRG simulations of the one-dimensional (1D) Hubbard model. In
particular, the particle density and its correlations show crossovers among
different regimes: for strongly repulsive interactions they approach those of a
spinless Fermi gas, for weak interactions those of a free Fermi gas, and for
strongly attractive interactions they match those of a gas of hard-core bosonic
molecules.
  The large-N behavior of systems at fixed N/l corresponds to a 1D trap
thermodynamic limit. We address issues related to the accuracy of the local
density approximation (LDA). We show that the particle density approaches its
LDA in the large-l limit. When the trapped system is in the metallic phase,
corrections at finite l are O(l^{-1}) and oscillating around the center of the
trap. They become significantly larger at the boundary of the fermion cloud,
where they get suppressed as O(l^{-1/3}) only. This anomalous behavior arises
from the nontrivial scaling at the metal-to-vacuum transition occurring at the
boundaries of the fermion cloud.
