2014/04/21 by Paulo Manrique, Manrique, Paulo, Vı́ctor Pérez-Abreu +3
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1404.5340
openalex publication_date 2014/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the universal asymptotically almost sure non-singularity of general\nGinibre and Wigner ensembles of random matrices when the distribution of the\nentries are independent but not necessarily identically distributed and may\ndepend on the size of the matrix. These models include adjacency matrices of\nrandom graphs and also sparse, generalized, universal and banded random\nmatrices. We find universal rates of convergence and precise estimates for the\nprobability of singularity which depend only on the size of the biggest jump of\nthe distribution functions governing the entries of the matrix and not on the\nrange of values of the random entries. Moreover, no moment assumptions are made\nabout the distributions governing the entries. Our proofs are based on a\nconcentration function inequality due to Kolmogorov, Rogozin and Kesten, which\nallows us to improve universal rates of convergence for the Wigner case when\nthe distribution of the entries do not depend on the size of the matrix.\n