Recursion for the Smallest Eigenvalue Density of β WishartLaguerre Ensemble
Abstract
The statistics of the smallest eigenvalue of WishartLaguerre ensemble is important from several perspectives. The smallest eigenvalue density is typically expressible in terms of determinants or Pfaffians. These results are of utmost significance in understanding the spectral behavior of WishartLaguerre ensembles and, among other things, unveil the underlying universality aspects in the asymptotic limits. However, obtaining exact and explicit expressions by expanding determinants or Pfaffians becomes impractical if large dimension matrices are involved. For the real matrices (β =1) Edelman has provided an efficient recurrence scheme to work out exact and explicit results for the smallest eigenvalue density which does not involve determinants or matrices. Very recently, an analogous recurrence scheme has been obtained for the complex matrices (β =2). In the present work we extend this to β WishartLaguerre ensembles for the case when exponent α in the associated Laguerre weight function, λ ^α e^{β λ /2}, is a nonnegative integer, while β is positive real. This also gives access to the smallest eigenvalue density of fixed trace β WishartLaguerre ensemble, as well as moments for both cases. Moreover, comparison with earlier results for the smallest eigenvalue density in terms of certain hypergeometric function of matrix argument results in an effective way of evaluating these explicitly. Exact evaluations for large values of n (the matrix dimension) and α also enable us to compare with TracyWidom density and large deviation results of Katzav and Castillo. We also use our result to obtain the density of the largest of the proper delay times which are eigenvalues of the WignerSmith matrix and are relevant to the problem of quantum chaotic scattering.
 Publication:

Journal of Statistical Physics
 Pub Date:
 April 2019
 DOI:
 10.1007/s1095501902245z
 arXiv:
 arXiv:1708.08646
 Bibcode:
 2019JSP...175..126K
 Keywords:

 Mathematical Physics;
 Condensed Matter  Statistical Mechanics;
 Statistics  Applications;
 15B52;
 15A18;
 65Q30;
 81Q50;
 33C70;
 60F10;
 81V65
 EPrint:
 Published version