Local law and Tracy–Widom limit for sparse random matrices

Abstract

We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős-Rényi graph model G(N, p). We prove a local law for the eigenvalue density up to the spectral edges. Under a suitable condition on the sparsity, we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included. For the adjacency matrix of the Erdős-Rényi graph this establishes the Tracy-Widom fluctuations of the second largest eigenvalue for p ≫ N −2/3 with a deterministic shift of order (N p) −1 .

Quasi-universality in single-cell sequencing data

“…We observe that this transition also occurs in the Wishart/covariance ensemble at some value of non-zero entries per row. In the case of medium sparsity (fraction of non-zero values p > N -1/3 , where N is the size of the matrix) it was recently reported(37)…”

Local law and Tracy–Widom limit for sparse stochastic block models

“…One of the most notable aspects of sparse random matrices is that the deterministic shift of their largest eigenvalues are much larger than the size of the Tracy-Widom fluctuation. Thus, as discussed in Remark 2.14 of [26], in cases where the intra-community probability p s and the inter-community probability p d are both small and close to each other, we can predict that the algorithms for the community detection should reflect the shift of the largest eigenvalues if p s , p d ≪ N −1/3 . However, to our best knowledge, it has not been proved for sparse SBM.…”

“…The second aspect is common in many real data, since the expected degree is much smaller than N and the edge probability decays as N grows. For sparse random matrices with identical off-diagonal entries, which correspond to Erdős-Rényi graphs, the spectral properties were obtained in [12,10,26]. One of the most notable aspects of sparse random matrices is that the deterministic shift of their largest eigenvalues are much larger than the size of the Tracy-Widom fluctuation.…”

“…Since the weak local law is not sufficient for the proof of other properties such as the Tracy-Widom fluctuation of the extremal eigenvalues, we improve it to prove the strong local law for the normalized trace of the resolvent (Theorem 2.6), which is optimal up to the edge of the spectrum, by adapting the strategy of [26,23].…”

Eigenvalue confinement and spectral gap for random simplicial complexes

“…[]). (iii) The local spectral statistics in the bulk spectrum are known to exhibit the universal GOE statistics from random matrix theory provided that for some (see ), and the second‐largest eigenvalue is known to exhibit the universal Tracy‐Widom statistics from random matrix theory provided that for some (see ). The goal of this paper is to extend (i) and (ii) above to the high‐dimensional setting d > 1.…”