2025/09/05 by Kartick Adhikari, Adhikari, Kartick, Samiron Parui +1 · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #Convergence of probability measures #FOS: Mathematics #Geometry and complex manifolds #Hypergraphs #Probability (math.PR) #Random Matrices and Applications #Random graphs(graph-theoretic aspects) #Random matrices (probabilistic aspects) #Topological and Geometric Data Analysis #Weak limit
paper · pdf · doi:10.48550/arxiv.2509.05102
openalex publication_date 2025/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≤ r≤ k-1, define the r-set weighted line graph Gr(H)=(\mathcal Vr,\mathcal Er,wH) by \mathcal Vr=[\tbinomnr], \mathcal Er=\\τ1,τ2\:τ1,τ2∈\mathcal Vr, ∃ e∈ E s.t. τ1,τ2⊂ e\, with weight wH(\τ1,τ2\)=|\e∈ E:τ1,τ2⊂ e\|. In particular, G1(Hn) generalizes Erd\H os--Rényi graphs and Gk-1(Hn) is the line graph of the Linial--Meshulam complex. We show that if \tbinomn-rk-r→ λ as n→∞, then Gr(Hn) converges locally to the (\tbinomkr-1)-block Galton--Watson measure with Poisson(λ) offspring almost surely. As a consequence, we obtain the limiting spectral distribution of the adjacency matrices of Gr(Hn).