Title: Recursion for the smallest eigenvalue density of $β$-Wishart-Laguerre ensemble,
Abstract: The statistics of the smallest eigenvalue of Wishart-Laguerre 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
Wishart-Laguerre 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 ($\beta=1$)
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 ($\beta=2$). In the present work we
extend this to $\beta$-Wishart-Laguerre ensembles for the case when exponent
$\alpha$ in the associated Laguerre weight function, $\lambda^\alpha
e^{-\beta\lambda/2}$, is a non-negative integer, while $\beta$ is positive
real. This also gives access to the smallest eigenvalue density of fixed trace
$\beta$-Wishart-Laguerre 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 $\alpha$ also enable us to compare with Tracy-Widom
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 Wigner-Smith matrix and are relevant to the problem of
quantum chaotic scattering.