Introduced in 1928, many years before the celebrated Gaussian ensembles of
Wigner-Dyson, the Wishart ensemble contains covariance matrices of a
maximally random data set.
The 'N' eigenvalues constitute a set of strongly correlated random
variables, for which a number of
analytical results (both for finite N and for N\to\infty) can be derived.
In the first part, I will present general results and techniques, and give
two examples of application:
1) spectral properties of financial data sets (and Marcenko-Pastur
distribution).
2) random pure entangled states in a bipartite Hilbert space (and
distribution of the smallest eigenvalue).
In the second part, I concentrate on a recent result (large deviation of
the maximum eigenvalue) and on some work in progress (superstatistical
Wishart-Laguerre ensemble). I also point out the differences between
spectral results on a global (macroscopic, non-universal), and local
(microscopic, universal) scale.