Renormalizing rectangles and other topics in random matrix theory

Research output: Contribution to journalArticlepeer-review


We consider random Hermitian matrices made of complex or real M × N rectangular blocks, where the blocks are drawn from various ensembles. These matrices have N pairs of opposite real nonvanishing eigenvalues, as well as M - N zero eigenvalues (for M > N). These zero eigenvalues are "kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the "rectangularity" r = M/N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large-N renormalization techniques. In addition to the kinematical δ-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r - 1| is held fixed as N → ∞, the N nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As r → 1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that r → 1 drives a crossover to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

Original languageEnglish
Pages (from-to)473-504
Number of pages32
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - May 1997
Externally publishedYes


  • Central limit theorem
  • Dirac operators in random backgrounds
  • Large-N limit
  • Large-N renormalization group
  • Random matrices with chiral block structure
  • Random matrix theory
  • Transport in disordered conductors

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


Dive into the research topics of 'Renormalizing rectangles and other topics in random matrix theory'. Together they form a unique fingerprint.

Cite this