Deformations of the Tracy-Widom distribution

O. Bohigas 1, J. X. de Carvalho 2, 3, M. P. Pato 1, 2

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 79 (2008) 031117

In random matrix theory (RMT), the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian ensembles by superimposing an external source of randomness. A competition between TW and a normal (Gaussian) distribution results, depending on the spreading of the disorder. The second model consists in removing at random a fraction of (correlated) eigenvalues of a random matrix. The usual formalism of Fredholm determinants extends naturally. A continuous transition from TW to the Weilbull distribution, characteristc of extreme values of an uncorrelated sequence, is obtained.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. Instituto de Fisica,
    Universidade de São Paulo
  • 3. Max-Planck-Institut für Physik komplexer Systeme,
    Max-Planck-Institut