ELSD in the fully sparse non-uniform case with two fixed hyperedge sizes

Determine the expected limiting spectral distribution of the centered-and-scaled adjacency matrices H_n = (A − E[A]) / sqrt(n σ^2) for the non-uniform inhomogeneous Erdős–Rényi hypergraph model H(n, r, p) with k = 2 and fixed hyperedge sizes r_1 and r_2, in the fully sparse regime where the average degrees d_1 and d_2 converge to finite positive limits (equivalently, p_i = Θ(n^{1 − r_i})). Provide an explicit description of the limiting probability measure, if it exists.

Background

The paper’s main results identify semicircle laws for the ELSD in non-sparse regimes and recover known results for uniform cases. For fixed hyperedge sizes, if at least one uniform component has diverging average degree, the ELSD is the standard semicircle law.

In uniform sparse hypergraphs with fixed r and average degree tending to a finite limit, the LSD is the distribution Γ_{r−1,λ}. The authors consider the non-uniform case with two fixed hyperedge sizes and summarize known and unknown regimes in a comparison table.

They explicitly point out that when both uniform subhypergraphs are sparse (both average degrees tend to finite limits), their theorem does not apply and the limiting spectral distribution in this regime is unknown.