Sparse Random Block Matrices : universality (original) (raw)
Related papers
Journal of Physics A: Mathematical and Theoretical, 2021
The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erdös-Renyi random graph. The blocks are i.i.d. random matrices of the classical ensembles GOE or GUE. The moments are evaluated for finite or infinite dimension of the blocks. The correspondences between sets of closed walks on trees and classes of irreducible partitions studied in free probability together with functional relations are powerful tools for analytic evaluation of the limiting moments. They are helpful to identify probability laws for the blocks and limits of the parameters which allow the evaluation of all the spectral moments and of the spectral density.
Spectral properties of random matrices for stochastic block model
2015 13th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), 2015
We consider an extension of Erdős-Rényi graph known in literature as Stochastic Block Model (SBM). We analyze the limiting empirical distribution of the eigenvalues of the adjacency matrix of SBM. We derive a fixed point equation for the Stieltjes transform of the limiting eigenvalue empirical distribution function (e.d.f.), concentration results on both the support of the limiting e.s.f. and the extremal eigenvalues outside the support of the limiting e.d.f. Additionally, we derive analogous results for the normalized Laplacian matrix and discuss potential applications of the general results in epidemics and random walks.
S ep 2 01 5 Index statistical properties of sparse random graphs
2015
Using the replica method, we develop an analytical approach to compute the characteristic function for the probability PN(K,λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of PN(K,λ), from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N ≫ 1 for |λ| > 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as lnN , with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support ...
Spectra of sparse random matrices
Journal of Physics A: Mathematical and Theoretical, 2008
We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by Rodgers and Bray. Due attention is payed to the issue of localization. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows to unfold the total density of states into contributions coming from vertices of different local coordination.
Limiting Spectral Distributions of Families of Block Matrix Ensembles
2021
We introduce a new matrix operation on a pair of matrices, swirl(A,X), and discuss its implications on the limiting spectral distribution. In a special case, the resultant ensemble converges almost surely to the Rayleigh distribution. In proving this, we provide a novel combinatorial proof that the random matrix ensemble of circulant Hankel matrices converges almost surely to the Rayleigh distribution, using the method of moments.
Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues
Communications in Mathematical Physics, 2012
We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN ≫ N 2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.
Spectral Analysis of Random Sparse Matrices
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2011
We study n × n random symmetric matrices whose entries above the diagonal are iid random variables each of which takes 1 with probability p and 0 with probability 1 − p, for a given density parameter p = α/n for sufficiently large α. For a given such matrix A, we consider a matrix A that is obtained by removing some rows and corresponding columns with too many value 1 entries. Then for this A , we show that the largest eigenvalue is asymptotically close to α + 1 and its eigenvector is almost parallel to all one vector (1, . . . , 1).
Proof of a Conjecture on the Infinite Dimension Limit of a Unifying Model for Random Matrix Theory
Journal of Statistical Physics, 2019
We study the large N limit of a sparse random block matrix ensemble. It depends on two parameters: the average connectivity Z and the size of the blocks d, which is the dimension of an euclidean space. In the limit of large d with Z d fixed, we prove the conjecture that the spectral distribution of the sparse random block matrix converges in the case of the Adjacency block matrix to the one of the effective medium approximation, in the case of the Laplacian block matrix to the Marchenko-Pastur distribution. We extend previous analytical computations of the moments of the spectral density of the Adjacency block matrix and the Lagrangian block matrix, valid for all values of Z and d.
Index statistical properties of sparse random graphs
Physical Review E, 2015
Using the replica method, we develop an analytical approach to compute the characteristic function for the probability PN (K, λ) that a large N × N adjacency matrix of sparse random graphs has K eigenvalues below a threshold λ. The method allows to determine, in principle, all moments of PN (K, λ), from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with N ≫ 1 for |λ| > 0, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erdös-Rényi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of λ. These results contrast with rotationally invariant random matrices, where the index variance scales only as ln N , with an universal prefactor that is independent of λ. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.