  We consider the discrete time dynamics of an ensemble of fermionic quantum
walkers moving on a finite discrete sample, interacting with a reservoir of
infinitely many quantum particles on the one dimensional lattice. The reservoir
is given by a fermionic quasifree state, with free discrete dynamics given by
the shift, whereas the free dynamics of the non-interacting quantum walkers in
the sample is defined by means of a unitary matrix. The reservoir and the
sample exchange particles at specific sites by a unitary coupling and we study
the discrete dynamics of the coupled system defined by the iteration of the
free discrete dynamics acting on the unitary coupling, in a variety of
situations. In particular, in absence of correlation within the particles of
the reservoir and under natural assumptions on the sample's dynamics, we prove
that the one- and two-body reduced density matrices of the sample admit large
times limits characterized by the state of the reservoir which are independent
of the free dynamics of the quantum walkers and of the coupling strength.
Moreover, the corresponding asymptotic density profile in the sample is flat
and the correlations of number operators have no structure, a manifestation of
thermalization.
