  We consider $m$ spinless Fermions in $l > m$ degenerate single-particle
levels interacting via a $k$-body random interaction with Gaussian probability
distribution and $k <= m$ in the limit $l$ to infinity (the embedded $k$-body
random ensembles). We address the cases of orthogonal and unitary symmetry. We
derive a novel eigenvalue expansion for the second moment of the Hilbert-space
matrix elements of these ensembles. Using properties of the expansion and the
supersymmetry technique, we show that for $2k > m$, the average spectrum has
the shape of a semicircle, and the spectral fluctuations are of Wigner-Dyson
type. Using a generalization of the binary correlation approximation, we show
that for $k << m << l$, the spectral fluctuations are Poissonian. This is
consistent with the case $k = 1$ which can be solved explicitly. We construct
limiting ensembles which are either fully integrable or fully chaotic and show
that the $k$-body random ensembles lie between these two extremes. Combining
all these results we find that the spectral correlations for the embedded
ensembles gradually change from Wigner-Dyson for $2k > m$ to Poissonian for $k
<< m << l$.
