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We study the statistics of quantum transmission through a one-dimensionaldisordered system modelled by a sequence of independent scattering units. Eachunit is characterized by its length and by its action, which is proportional tothe logarithm of the transmission probability through this unit. Unit actionsand lengths are independent random variables, with a common distribution thatis either narrow or broad. This investigation is motivated by results ondisordered systems with non-stationary random potentials whose fluctuationsgrow with distance.  In the statistical ensemble at fixed total sample length four phases can bedistinguished, according to the values of the indices characterizing thedistribution of the unit actions and lengths. The sample action, which isproportional to the logarithm of the conductance across the sample, is found toobey a fluctuating scaling law, and therefore to be non-self-averaging, inthree of the four phases. According to the values of the two above mentionedindices, the sample action may typically grow less rapidly than linearly withthe sample length (underlocalization), more rapidly than linearly(superlocalization), or linearly but with non-trivial sample-to-samplefluctuations (fluctuating localization).

Statistics of quantum transmission in one dimension with broad disorder