In disordered Hermitian systems, localization of energy eigenstates prohibits wave propagation. In non-Hermitian systems, however, wave propagation is possible even when the eigenstates of a Hamiltonian are exponentially localized by disorders. We find in this regime that non-Hermitian wave propagation exhibits novel universal scaling behaviors without Hermitian counterpart. Furthermore, our theory demonstrates how the tail of imaginary-part density of states dictates wave propagation in the long-time limit. Specifically, for the three typical classes, namely the Gaussian, the uniform, and the linear imaginary-part density of states, we obtain logarithmically suppressed sub-ballistic transport, and two types of subdiffusion with exponents that depend only on spatial dimensions, respectively. Our work highlights the fundamental differences between Hermitian and non-Hermitian Anderson localization, and uncovers unique universality in non-Hermitian wave propagation.

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• Open quantum systems & decoherence
• PT-symmetric quantum mechanics
• Non-Hermitian systems