Extreme gaps between eigenvalues of random matrices

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Abstract

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth smallest gap, normalized by a factor n -4/3, has a limiting density proportional to x3k-1e -x3. Concerning the largest gaps, normalized by n/v log n, they converge in Lp to a constant for allp >0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

Original languageEnglish (US)
Pages (from-to)2648-2681
Number of pages34
JournalAnnals of Probability
Volume41
Issue number4
DOIs
StatePublished - Aug 20 2013

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Keywords

  • Eigenvalues statistics
  • Extreme spacings
  • Gaussian unitary ensemble
  • Negative association property
  • Random matrices

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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