The circular law for sparse non-Hermitian matrices
| Autor: | Anirban Basak, Mark Rudelson |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
sparse matrix
Statistics and Probability 60B10 smallest singular value 01 natural sciences Omega circular law Combinatorics 010104 statistics & probability FOS: Mathematics Almost surely 0101 mathematics Mathematics 60B20 010102 general mathematics Probability (math.PR) Random matrix 15B52 Hermitian matrix Circular law 15B52 60B10 60B20 Singular value Statistics Probability and Uncertainty Random variable Unit (ring theory) Mathematics - Probability |
| Zdroj: | Ann. Probab. 47, no. 4 (2019), 2359-2416 |
| DOI: | 10.48550/arxiv.1707.03675 |
| Popis: | For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, we prove that the empirical spectral distribution of $A_n/\sqrt{np_n}$ converges weakly to the circular law, in probability, for all $p_n$ such that $p_n=\omega({\log^2n}/{n})$. Additionally if $p_n$ satisfies the inequality $np_n > \exp(c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erd\H{o}s-R\'{e}nyi graph with edge connectivity probability $p_n$. Comment: 55 pages, Section 9 shortened, presentation improved, proof of Theorem 1.7 is removed from this version. For its proof we refer the reader to V1 |
| Databáze: | OpenAIRE |
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