The sparse circular law under minimal assumptions

  • Mark RudelsonEmail author
  • Konstantin Tikhomirov


The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized \({n \times n}\) matrix with i.i.d. entries converges to the uniform measure on the unit disc as the dimension n grows to infinity. Consider an \({n \times n}\) matrix \({A_n=(\delta_{ij}^{(n)}\xi_{ij}^{(n)})}\), where \({\xi_{ij}^{(n)}}\) are copies of a real random variable of unit variance, variables \({\delta_{ij}^{(n)}}\) are Bernoulli (0/1) with \({\mathbb{P}\{\delta_{ij}^{(n)} = 1\} = p_n}\), and \({\delta_{ij}^{(n)}}\) and \({\xi_{ij}^{(n)}, i, j \in [n]}\), are jointly independent. In order for the circular law to hold for the sequence \({\big(\frac{1}{\sqrt{p_{n}n}}A_{n}\big)}\), one has to assume that \({p_{n}n \to \infty}\). We derive the circular law under this minimal assumption.


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Part of this research was performed while the authors were in residence at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California, during the Fall semester of 2017, and at the Institute for Pure and Applied Mathematics (IPAM) in Los Angeles, California, during May and June of 2018. Both institutions are supported by the National Science Foundation. Part of this research was performed while the first author visited Weizmann Institute of Science in Rehovot, Israel, where he held Rosy and Max Varon Professorship. We are grateful to all these institutes for their hospitality and for creating an excellent work environment. The research of the first author was supported in part by the NSF Grant DMS 1464514 and by a fellowship from the Simons Foundation. The research of the second named author was supported by the Viterbi postdoctoral fellowship while in residence at the Mathematical Sciences Research Institute.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.Georgia Institute of TechnologyAtlantaUSA

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