Proceedings of the National Academy of Sciences (PNAS) **111**, 2926-2930 (2014).

The nature of transport of electrons and classical waves in disordered systems depends upon the proximity to the Anderson localization transition between freely diffusing and localized waves. The suppression of average transport and the enhancement of relative fluctuations in conductance in one-dimensional samples with lengths greatly exceeding the localization length, L>>ξ, are related in the single-parameter scaling (SPS) theory of localization [1]. However, the difficulty of producing a collections of statistically equivalent random sample in which the electron wave function is temporally coherent has precluded the experimental demonstration of SPS. We have demonstrates SPS in random multichannel systems for the transmittance of microwave radiation, which is the analog of the electronic dimensionless conductance. We show that for L∼4ξ, a single eigenvalue of the transmission matrix (TM) dominates transmission, and the distribution of the logarithm of the transmittance, ln T is Gaussian with a variance equal to the average of −ln T, as conjectured by SPS. For samples in the cross-over to localization, L∼ξ, we find a one-sided distribution for ln T. This anomalous distribution is explained in terms of a charge model for the eigenvalues of the transmission matrix in which the Coulomb interaction between charges mimics the repulsion between the eigenvalues of transmission matrix. We show in the localization limit that the joint distribution of T and the effective number of transmission eigenvalues determines the probability distributions of intensity and total transmission for a single-incident channel.

1. Anderson PW, Thouless DJ, Abrahams E, Fisher DS (1980) New method for a scaling theory of localization. Phys Rev B 22:3519–3526.