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The emergence of nonequilibrium phenomena in individual complex wave systemshas long been of fundamental interests. Its analytic studies remain notoriouslydifficult. Using the mathematical tool of the concentration of measure (CM), wedevelop a theory for structures and fluctuations of waves in individualdisordered media. We find that, for both diffusive and localized waves,fluctuations associated with the change in incoming waves ("wave-to-wave"fluctuations) exhibit a new kind of universalities, which does not exist inconventional mesoscopic fluctuations associated with the change in disorderrealizations ("sample-to-sample" fluctuations), and originate from thecoherence between the natural channels of waves -- the transmissioneigenchannels. Using the results obtained for wave-to-wave fluctuations, wefind the criterion for almost all stationary scattering states to exhibit thesame spatial structure such as the diffusive steady state. We further show thatthe expectations of observables at stationary scattering states are independentof incoming waves and given by their averages with respect to eigenchannels.This suggests the possibility of extending the studies of thermalization ofclosed systems to open systems, which provides new perspectives for theemergence of nonequilibrium statistical phenomena.

Concentration-of-measure theory for structures and fluctuations of waves