  We consider the adjacency operator of the Linial-Meshulam model for random
simplicial complexes on $n$ vertices, where each $d$-cell is added
independently with probability $p$ to the complete $(d-1)$-skeleton. Under the
assumption $np(1-p) \gg \log^4 n$, we prove that the spectral gap between the
$\binom{n-1}{d}$ smallest eigenvalues and the remaining $\binom{n-1}{d-1}$
eigenvalues is $np - 2\sqrt{dnp(1-p)} \, (1 + o(1))$ with high probability.
This estimate follows from a more general result on eigenvalue confinement. In
addition, we prove that the global distribution of the eigenvalues is
asymptotically given by the semicircle law. The main ingredient of the proof is
a F\"uredi-Koml\'os-type argument for random simplicial complexes, which may be
regarded as sparse random matrix models with dependent entries.
