# The sparse circular law under minimal assumptions

@article{Rudelson2019TheSC,
title={The sparse circular law under minimal assumptions},
author={Mark Rudelson and Konstantin E. Tikhomirov},
journal={Geometric and Functional Analysis},
year={2019},
volume={29},
pages={561-637}
}
• Published 21 July 2018
• Mathematics
• Geometric and Functional Analysis
The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $${n \times n}$$n×n matrix with i.i.d. entries converges to the uniform measure on the unit disc as the dimension n grows to infinity. Consider an $${n \times n}$$n×n matrix $${A_n=(\delta_{ij}^{(n)}\xi_{ij}^{(n)})}$$An=(δij(n)ξij(n)), where $${\xi_{ij}^{(n)}}$$ξij(n) are copies of a real random variable of unit variance, variables $${\delta_{ij}^{(n)}}$$δij(n) are Bernoulli (0/1) with $${\mathbb… 9 Citations Small Ball Probability for the Condition Number of Random Matrices • Mathematics • 2020 Let A be an n × n random matrix with i.i.d. entries of zero mean, unit variance and a bounded sub-Gaussian moment. We show that the condition number $$s_{\max }(A)/s_{\min }(A)$$ satisfies the small Sharp transition of the invertibility of the adjacency matrices of sparse random graphs • Mathematics • 2018 We consider three different models of sparse random graphs:~undirected and directed Erdős-Renyi graphs, and random bipartite graph with an equal number of left and right vertices. For such graphs we Circular law for random block band matrices with genuinely sublinear bandwidth • Mathematics, Physics • 2020 We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists \tau \in (0,1) so that if the bandwidth of the On words of non-Hermitian random matrices • Mathematics The Annals of Probability • 2019 We consider words G_{i_1} \cdots G_{i_m} involving i.i.d. complex Ginibre matrices, and study their singular values and eigenvalues. We show that the limit distribution of the squared singular Sparse matrices: convergence of the characteristic polynomial seen from infinity We prove that the reverse characteristic polynomial det(In − zAn) of a random n×nmatrixAn with iidBernoulli(d/n) entries converges in distribution towards the random infinite product ∞ ∏ =1 (1− z) Local and global geometry of the 2D Ising interface in critical prewetting • Mathematics • 2021 Consider the Ising model at low temperatures and positive external field λ on an N×N box with Dobrushin boundary conditions that are plus on the north, east and west boundaries and minus on the south Tail bounds for gaps between eigenvalues of sparse random matrices • Mathematics • 2019 We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability Convergence of the spectral radius of a random matrix through its characteristic polynomial • Mathematics Probability Theory and Related Fields • 2021 Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent Spectral radius of random matrices with independent entries • Probability and Mathematical Physics • 2021 ## References SHOWING 1-10 OF 43 REFERENCES The circular law for sparse non-Hermitian matrices • Mathematics The Annals of Probability • 2019 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 Invertibility of Sparse non-Hermitian matrices • Mathematics • 2015 We consider 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 random variables, and \{\delta_{i,j}\} are Circular law for the sum of random permutation matrices • Mathematics • 2017 Let P_n^1,\dots, P_n^d be n\times n permutation matrices drawn independently and uniformly at random, and set S_n^d:=\sum_{\ell=1}^d P_n^\ell. We show that if \log^{12}n/(\log \log n)^{4} \le The smallest singular value of a shifted d-regular random square matrix • Mathematics Probability Theory and Related Fields • 2018 We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let$$C_1<d< c n/\log ^2
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