  We study the details of the distribution of the entanglement spectrum
(eigenvalues of the reduced density matrix) of a disordered spin chain
exhibiting a many-body localization (MBL) transition. In the thermalizing
region we identify the evolution under increasing system size of the
eigenvalues distribution function, whose thermodynamic limit is close (but
possibly different from) the Marchenko-Pastur distribution. From the analysis
we extract a correlation length $L_s(h)$ determining the minimum system size to
enter the asymptotic region. We find that $L_s(h)$ diverges at the MBL
transition. We discuss the nature of the subleading corrections to the
entanglement spectrum distribution and to the entanglement entropy.
