On the condition number of the critically-scaled laguerre unitary ensemble

Percy A. Deift, Thomas Trogdon, Govind Menon

Research output: Contribution to journalArticlepeer-review


We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices A = XX, where X is an Nxn matrix with iid standard complex normal entries. Under the scaling n = N+ √4cN®90, c > 0 and N → ∞, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices A are all given by the Tracy-Widom distribution (β = 2). This scaling is motivated by the study of the solution of the equation Ax = b using the conjugate gradient algorithm, in the case that A and b are random: For such a scaling the fluctuations of the halting time for the algorithm are empirically seen to be universal.

Original languageEnglish (US)
Pages (from-to)4287-4347
Number of pages61
JournalDiscrete and Continuous Dynamical Systems- Series A
Issue number8
StatePublished - Aug 2016


  • Conjugate gradient algorithm
  • Laguerre polynomials
  • Laguerre unitary ensemble
  • Riemann-Hilbert problems
  • Wishart ensemble

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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