Loose Laplacian spectra of random hypergraphs

Let $H=(V,E)$ be an $r$-uniform hypergraph with the vertex set $V$ and the edge set $E$. For $1\le s\le r\mathrm{/}2$, we define a weighted graph $G^{(s)}$ on the vertex set $\left(\genfrac{}{}{0px}{}{V}{s}\right)$ 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 $H$ passing through $I$ and $J$ if $I\cap J=\mathrm{\varnothing }$, and 0 if $I\cap J≠\mathrm{\varnothing }$. The $s$-th Laplacian ${\text{L}}^{(s)}$ of $H$ is defined to be the normalized Laplacian of $G^{(s)}$. The eigenvalues of ${\mathcal{L}}^{(s)}$ are listed as ${\lambda }_0^{(s)},{\lambda }_1^{(s)},\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_{\left(\genfrac{}{}{0px}{}{n}{s}\right)-1}^{(s)}$ in non-decreasing order. Let ${\bar{\lambda }}^{(s)}(H)={\max }_{i≠0}\{\mathrm{\mid }1-{\lambda }_i^{(s)}\mathrm{\mid }\}$. The parameters ${\bar{\lambda }}^{(s)}(H)$ and ${\lambda }_1^{(s)}(H)$, 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 $[n]:=1,2,\mathrm{.}\mathrm{.}\mathrm{.},n$, where each $r$-set of $[n]$ has probability $p$ to be an edge independently. For $1\le s\le r\mathrm{/}2$, $p(1-p)\gg \frac{\log^4 n}{n^{r-s}}$,and$1-p\gg \frac{\log n}{n^2}$,weprovethatalmostsurely$$\bar\lambda^{(s)}(H^r(n,p))\leq \frac{s}{n-s}+ (3+o(1))\sqrt{\frac{1-p}{{n-s\choose r-s}p}}.$$ We also prove that the empirical distribution of the eigenvalues of ${\text{L}}^{(s)}$ for $H^r(n,p)$ follows the Semicircle Law if $p(1-p)\gg \frac{{\log }^{1\mathrm{/}3}n}{n^{r-s}}$ and $1-p\gg \frac{\log n}{n^{2+2r-2s}}$.