  We consider arrays of Luttinger liquids, where each node is described by a
unitary scattering matrix. In the limit of small electron-electron interaction,
we study the evolution of these scattering matrices as the high-energy single
particle states are gradually integrated out. Interestingly, we obtain the same
renormalization group equations as those derived by Lal, Rao, and Sen, for a
system composed of a single node coupled to several semi-infinite 1D wires. The
main difference between the single node geometry and a regular lattice is that
in the latter case, the single particle spectrum is organized into periodic
energy bands, so that the renormalization procedure has to stop when the last
totally occupied band has been eliminated. We therefore predict a strongly
renormalized Luttinger liquid behavior for generic filling factors, which
should exhibit power-law suppression of the conductivity at low temperatures
E_{F}/(k_{F}a) << k_{B}T << E_{F}, where a is the lattice spacing and k_{F}a >>
1. Some fully insulating ground-states are expected only for a discrete set of
integer filling factors for the electronic system. A detailed discussion of the
scattering matrix flow and its implication for the low energy band structure is
given on the example of a square lattice.
