  The present paper is devoted to the study of a simple model of interacting
electrons in a random background. In a large interval $\Lambda$, we consider
$n$ one dimensional particles whose evolution is driven by the Luttinger-Sy
model, i.e., the interval $\Lambda$ is split into pieces delimited by the
points of a Poisson process of intensity $\mu$ and, in each piece, the
Hamiltonian is the Dirichlet Laplacian. The particles interact through a
repulsive pair potential decaying polynomially fast at infinity. We assume that
the particles have a positive density, i.e., $n/|\Lambda|\to\rho>0$ as
$|\Lambda|\to+\infty$. In the low density or large disorder regime, i.e.,
$\rho/\mu$ small, we obtain a two term asymptotic for the thermodynamic limit
of the ground state energy per particle of the interacting system; the first
order correction term to the non interacting ground state energy per particle
is controlled by pairs of particles living in the same piece. The ground state
is described in terms of its one and two-particles reduced density matrix.
Comparing the interacting and the non interacting ground states, one sees that
the effect of the repulsive interactions is to move a certain number of
particles living together with another particle in a single piece to a new
piece that was free of particles in the non interacting ground state.
