Spectrum and local weak convergence of sparse random uniform hypergraphs

The notion of local weak convergence, or Benjamini–Schramm convergence, was introduced by Benjamini and Schramm. The local weak limit of sparse Erd\H os–Rényi graphs is the Galton–Watson measure with Poisson offspring almost surely. Recently, Adhikari, Kumar, and Saha showed that the line graph of sparse Linial–Meshulam complexes converges to the $d$-block Galton–Watson measure. We study a unified model: weighted line graphs of sparse $k$-uniform random hypergraphs on $n$ vertices. Let $H(n,k,p)$ be the $k$-uniform random hypergraph where each $k$-subset of $[n]$ is included as a hyperedge independently with probability $p$. For a $k$-uniform hypergraph $H=(V,E)$ and $1\le r\le k-1$, define the $r$-set weighted line graph $G_r(H)=({\mathcal{V}}_r,{\mathcal{E}}_r,w_H)$ by $${\mathcal{V}}_r=\left[\left(\genfrac{}{}{0px}{}{n}{r}\right)\right],{\mathcal{E}}_r=\{\{{\tau }_1,{\tau }_2\}:{\tau }_1,{\tau }_2\in {\mathcal{V}}_r,\mathrm{\exists }e\in E\text{ s.t. }{\tau }_1,{\tau }_2\subset e\},$$ with weight $w_H(\{{\tau }_1,{\tau }_2\})=\mid \{e\in E:{\tau }_1,{\tau }_2\subset e\}\mid$. In particular, $G_1(H_n)$ generalizes Erd\H os–Rényi graphs and $G_{k-1}(H_n)$ is the line graph of the Linial–Meshulam complex. We show that if $\left(\genfrac{}{}{0px}{}{n-r}{k-r}\right)\to \lambda$ as $n\to \mathrm{\infty }$, then $G_r(H_n)$ converges locally to the $(\left(\genfrac{}{}{0px}{}{k}{r}\right)-1)$-block Galton–Watson measure with $\mathrm{Poisson}(\lambda )$ offspring almost surely. As a consequence, we obtain the limiting spectral distribution of the adjacency matrices of $G_r(H_n)$.