  The competition between non-Hermitian skin effect and Anderson localization
leads to various intriguing phenomena concerning spectrums and wavefunctions.
Here, we study the linear response of disordered non-Hermitian systems, which
is precisely described by the Green's function. We find that the average
maximum value of matrix elements of Green's functions, which quantifies the
maximum response against an external perturbation, exhibits different phases
characterized by different scaling behaviors with respect to the system size.
Whereas the exponential-growth phase is also seen in the translation-invariant
systems, the algebraic-growth phase is unique to disordered non-Hermitian
systems. We explain the findings using the large deviation theory, which
provides analytical insights into the algebraic scaling factors of
non-Hermitian disordered Green's functions. Furthermore, we show that these
scaling behaviors can be observed in the steady states of disordered open
quantum systems, offering a quantum-mechanical avenue for their experimental
detection. Our work highlights an unexpected interplay between non-Hermitian
skin effect and Anderson localization.
