  We derive analytically the full distribution of the ground-state energy of
$K$ non-interacting fermions in a disordered environment, modelled by a
Hamiltonian whose spectrum consists of $N$ i.i.d.~random energy levels with
distribution $p(\varepsilon)$ (with $\varepsilon \geq 0$), in the same spirit
as the `Random Energy Model'. We show that for each fixed $K$, the distribution
$P_{K,N}(E_0)$ of the ground-state energy $E_0$ has a universal scaling form in
the limit of large $N$. We compute this universal scaling function and show
that it depends only on $K$ and the exponent $\alpha$ characterizing the small
$\varepsilon$ behaviour of $p(\varepsilon) \sim \varepsilon^\alpha$. We
compared the analytical predictions with results from numerical simulations.
For this purpose we employed a sophisticated importance-sampling algorithm that
allowed us to obtain the distributions over a large range of the support down
to probabilities as small as $10^{-160}$. We found asymptotically a very good
agreement between analytical predictions and numerical results.
