  The statistical behaviour of the smallest eigenvalue has important
implications for systems which can be modeled using a Wishart-Laguerre
ensemble, the regular one or the fixed trace one. For example, the density of
the smallest eigenvalue of the Wishart-Laguerre ensemble plays a crucial role
in characterizing multiple channel telecommunication systems. Similarly, in the
quantum entanglement problem, the smallest eigenvalue of the fixed trace
ensemble carries information regarding the nature of entanglement.
  For real Wishart-Laguerre matrices, there exists an elegant recurrence scheme
suggested by Edelman to directly obtain the exact expression for the smallest
eigenvalue density. In the case of complex Wishart-Laguerre matrices, for
finding exact and explicit expressions for the smallest eigenvalue density,
existing results based on determinants become impractical when the determinants
involve large-size matrices. In this work, we derive a recurrence scheme for
the complex case which is analogous to that of Edelman's for the real case.
This is used to obtain exact results for the smallest eigenvalue density for
both the regular, and the fixed trace complex Wishart-Laguerre ensembles. We
validate our analytical results using Monte Carlo simulations. We also study
scaled Wishart-Laguerre ensemble and investigate its efficacy in approximating
the fixed-trace ensemble. Eventually, we apply our result for the fixed-trace
ensemble to investigate the behaviour of the smallest eigenvalue in the
paradigmatic system of coupled kicked tops.
