  We consider $m$ spinless Bosons distributed over $l$ degenerate
single-particle states and interacting through a $k$-body random interaction
with Gaussian probability distribution (the Bosonic embedded $k$-body
ensembles). We address the cases of orthogonal and unitary symmetry in the
limit of infinite matrix dimension, attained either as $l \to \infty$ or as $m
\to \infty$. We derive an eigenvalue expansion for the second moment of the
many-body matrix elements of these ensembles. Using properties of this
expansion, the supersymmetry technique, and the binary correlation method, we
show that in the limit $l \to \infty$ the ensembles have nearly the same
spectral properties as the corresponding Fermionic embedded ensembles. Novel
features specific for Bosons arise in the dense limit defined as $m \to \infty$
with both $k$ and $l$ fixed. Here we show that the ensemble is not ergodic, and
that the spectral fluctuations are not of Wigner-Dyson type. We present
numerical results for the dense limit using both ensemble unfolding and
spectral unfolding. These differ strongly, demonstrating the lack of ergodicity
of the ensemble. Spectral unfolding shows a strong tendency towards
picket-fence type spectra. Certain eigenfunctions of individual realizations of
the ensemble display Fock-space localization.
