Eigenvalue confinement and spectral gap for random simplicial complexes

Understand

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.