Invariant pseudo-unitary matrix ensembles and local statistics and conductance fluctuations of the DMPK ensemble

Abstract

We consider the DMPK ensemble, which models quantum transport in a disordered wire, and describe its limiting local statistics as the number of particles tends to infinity. In the metallic regime, as well as at crossover I, we obtain sine statistics in the bulk, Bessel statistics at the hard edge and Airy statistics at the soft edge. In the insulating regime, we prove localization around certain explicit deterministic positions with Gaussian fluctuations. This confirms and quantifies the predicted universal and localized behavior in the physics literature. At crossover II, we obtain explicit limiting kernels in the bulk and at both edges, which interpolate between these regimes. The hard edge kernel involves Legendre functions and, to the best of our knowledge, is new, while the bulk and soft edge kernels agree with those derived for the squared singular values of critically scaled products of independent Ginibre matrices. To explain these similarities, we construct a class of invariant pseudo-unitary matrices that is closed under taking products and use it to interpret the DMPK ensemble as the joint density of the spectral variables of a product of such random matrices. More generally, the latter takes the form of a polynomial ensemble of derivative type involving a second order differential operator compatible with the Legendre functions. Finally, we obtain uniform asymptotic expansions of the mean and variance of the conductance and prove that the DMPK ensemble admits universal conductance fluctuations, i.e. the variance of the conductance tends to $1/15$ throughout the whole metallic regime. This again confirms the predictions in the physics literature.