Local weak limit of the non-uniform Erdős–Rényi hypergraph in the sparse regime

Establish the Benjamini–Schramm local weak limit of the non-uniform inhomogeneous Erdős–Rényi hypergraph model H(n, r, p) defined in Definition 2.1, allowing r = (r_i) and p = (p_i) to depend on n, in the sparse regime where local weak convergence techniques are required. Characterize the limiting rooted random hypergraph to enable identification of the limiting spectral distribution in the sparse case.

Background

The paper analyzes the expected limiting spectral distribution (ELSD) for centered-and-scaled adjacency matrices of non-uniform, inhomogeneous Erdős–Rényi hypergraphs in non-sparse regimes via a Gaussianization approach leading to semicircle laws.

For sparse models, the authors note that identifying the limiting spectral distribution typically relies on local weak convergence methods à la Benjamini–Schramm, as developed and applied by Bordenave and Lelarge to graphs and by Adhikari and Parui to uniform hypergraphs.

While these tools are available, the local weak limit of the present non-uniform model H(n, r, p) has not yet been derived, and the authors explicitly defer this analysis, which is necessary to tackle the sparse regime.