Large deviations of the top eigenvalue of large Cauchy random matrices

We compute analytically the large deviation tails of the probability density function (pdf) of the top eigenvalue λmax  in rotationally invariant and heavy-tailed Cauchy ensembles of N × N matrices for any Dyson index β > 0, where β = 1, 2, 4 correspond, respectively, to orthogonal, unitary and symplectic ensembles. Furthermore, we show that these large deviation tails flank a central non-Gaussian regime for on both sides. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors for any β of the pdf in the central regime, which generalizes the Tracy–Widom distribution known for Gaussian ensembles. Our analytical results are confirmed by numerical simulations.

• Publication year

  2012

• Venue

  Journal of Physics A: Mathematical and Theoretical

• Publication date

  2012-10-19

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1088/1751-8113/46/2/022001arXiv 1210.5400
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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