  We consider the distribution of the transmission coefficients, i.e. the
singular values of the modal transmission matrix, for 2D random media with
periodic boundary conditions composed of a large number of point-like
non-absorbing scatterers. The scatterers are placed at random locations in the
medium and have random refractive indices that are drawn from an arbitrary,
known distribution. We construct a randomized model for the scattering matrix
that retains scatterer dependent properties essential to reproduce the
transmission coefficient distribution and analytically characterize the
distribution of this matrix as a function of the refractive index distribution,
the number of modes, and the number of scatterers. We show that the derived
distribution agrees remarkably well with results obtained using a numerically
rigorous spectrally accurate simulation. Analysis of the derived distribution
provides the strongest principled justification yet of why we should expect
perfect transmission in such random media regardless of the refractive index
distribution of the constituent scatterers. The analysis suggests a sparsity
condition under which random media will exhibit a perfect
transmission-supporting universal transmission coefficient distribution in the
deep medium limit.
