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• W2962804035 abstract "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 $p_{n}=omega({log^{2}n}/{n})$. Additionally if $p_{n}$ satisfies the inequality $np_{n}>exp(csqrt{log n})$ 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 Erdős–Rényi graph with edge connectivity probability $p_{n}$." @default.
• W2962804035 created "2019-07-30" @default.
• W2962804035 creator A5057215536 @default.
• W2962804035 creator A5090571169 @default.
• W2962804035 date "2019-07-01" @default.
• W2962804035 modified "2023-10-09" @default.
• W2962804035 title "The circular law for sparse non-Hermitian matrices" @default.
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• W2962804035 doi "https://doi.org/10.1214/18-aop1310" @default.
• W2962804035 hasPublicationYear "2019" @default.
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