2024/09/05 by Mukherjee, Soumendu Sundar, Pal, Dipranjan, Talukdar, Himasish · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2409.03756
We study adjacency and Laplacian matrices of Erdős-Rényi r-uniform hypergraphs on n vertices with hyperedge inclusion probability p, in the setting where r can vary with n such that r / n → c ∈ [0, 1). Adjacency matrices of hypergraphs are contractions of adjacency tensors and their entries exhibit long range correlations. We show that under the Erdős-Rényi model, the expected empirical spectral distribution of an appropriately normalised hypergraph adjacency matrix converges weakly to the semi-circle law with variance (1 - c)2 as long as \fracd\avgr7 → ∞, where d\avg = \binomn-1r-1 p. In contrast with the Erdős-Rényi random graph (r = 2), two eigenvalues stick out of the bulk of the spectrum. When r is fixed and d\avg ≫ nr - 2 log4 n, we uncover an interesting Baik-Ben Arous-Péché (BBP) phase transition at the value r = 3. For r ∈ \2, 3\, an appropriately scaled largest (resp. smallest) eigenvalue converges in probability to 2 (resp. -2), the right (resp. left) end point of the support of the standard semi-circle law, and when r ≥ 4, it converges to √(r - 2) + (1)/(√(r - 2)) (resp. -√(r - 2) - (1)/(√(r - 2))). Further, in a Gaussian version of the model we show that an appropriately scaled largest (resp. smallest) eigenvalue converges in distribution to (c)/(2) ζ+ [(c2)/(4)ζ2 + c(1 - c)]1/2 (resp. (c)/(2) ζ- [(c2)/(4)ζ2 + c(1 - c)]1/2), where ζ is a standard Gaussian. We also establish analogous results for the bulk and edge eigenvalues of the associated Laplacian matrices.