A note on pair-dependent linear statistics with a slowly increasing variance

Abstract

We prove Gaussian fluctuations for pair-counting statistics of the form \(\Sigma_{1\le i\ne j\le N}f(\theta_i-\theta_j)\) for the circular unitary ensemble of random matrices in the large-\(N\) limit under the condition that the variance increases slowly as \(N\) increases.

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Funding

This research was supported in part by the Simons Foundation (Collaboration Grant for Mathematicians No. 312391).

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Correspondence to A. B. Soshnikov.

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Aguirre, A., Soshnikov, A.B. A note on pair-dependent linear statistics with a slowly increasing variance. Theor Math Phys 205, 1682–1691 (2020). https://doi.org/10.1134/S0040577920120090

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Keywords

  • random matrix
  • circular unitary ensemble
  • pair-counting statistics
  • central limit theorem