Summary: Let \(H = (V, E)\) be an \(r\)-uniform hypergraph with the vertex set \(V\) and the edge set \(E\). For \(1 \leq s \leq r/2\), we define a weighted graph \(G^{(s)}\) on the vertex set \(V \choose s\) as follows. Every pair of \(s\)-sets \(I\) and \(J\) is associated with a weight \(w(I, J)\), which is the number of edges in \(H\) containing \(I\) and \(J\) if \(I\cap J = \emptyset\), and 0 if \(I\cap J\neq \emptyset\). The \(s\)-th Laplacian \(\mathcal L^{(s)}\) of \(H\) is defined to be the normalized Laplacian of \(G^{(s)}\). The eigenvalues of \(\mathcal L^{(s)}\) are listed as \(\lambda^{(s)}_{0},\lambda^{(s)}_{1},\dots,\lambda^{(s)}_{{n\choose s}-1}\) in non-decreasing order.
Let \(\overline{\lambda}^{(s)}(H)=\)max\(_{i\neq 0}\{|1-\lambda^{(s)}_{i}|\}\). The parameters \(\overline{\lambda}^{(s)}(H)\) and \(\overline{\lambda}^{(s)}_1(H)\), which were introduced in our previous paper, have a number of connections to the mixing rate of high-ordered random walks, the generalized distances/diameters, and the edge expansions.
For \(0 < p < 1\), let \(H^{r}(n, p)\) be a random \(r\)-uniform hypergraph over \([n] := \{1, 2, \ldots , n\}\), where each \(r\)-set of \([n]\) has probability \(p\) to be an edge independently. For \(1 \leq s\leq r/2\), \(p(1-p)\gg \frac {\log^{4}n}{n^{r-s}}\), and \(p(1-p)\gg\frac {\log n}{n^2}\), we prove that almost surely
\[ \overline{\lambda}^{(s)}\left(H^{r}(n,p)\right) \leq {\frac {s}{n-s}+(3+o(1))\sqrt{\frac {1-p}{{n-s \choose {r-s}}p}}} \]
We also prove that the empirical distribution of the eigenvalues of \(\mathcal L^{(s)}\) for \(H^{r}(n, p)\) follows the semicircle law if \(p(1-p)\gg\frac {\log^{1/3}n}{n^{r-s}}\) and \(1-p\gg \frac {\log n}{n^{2+2r-2s}}\).