  We develop a general theory to study strong random quenched disorder effects
in systems of experimental relevance in the search for Majorana zero modes
(MZM) in topological superconductors. Using the random matrix theory in a class
D ensemble, we simulate the transport properties of random quantum dots by
attaching leads, and calculating the differential conductance in the $S$ matrix
formalism. To add the concept of the length to the random system so that
disordered Majorana nanowires can be simulated by the random matrix theory, we
generalize the model of a single quantum dot to a chain of quantum dots by
analogy with the superconductor-semiconductor (SC-SM) nanowire Majorana
platform. We first define a new concept, the robustness of zero-bias
conductance peaks (ZBCPs), in terms of an effective random Hamiltonian
considering the self-energy of leads. We then study the joint distribution for
the robustness and ZBCPs, and find a strong correlation that the ZBCP with
stronger robustness is also prone to carry a larger conductance peak near
$2e^2/h$. This trend is more prominent in shorter chains than in longer chains.
This is consistent with experimentally observed ZBCPs associated with
disorder-induced trivial Andreev bound states (the so-called ugly ZBCPs).
Finally, we study the end-to-end correlation of the disorder-induced ZBCPs from
two leads by calculating the normalized mutual information, which estimates the
degrees of the correlation arising from the trivial ZBCPs. Our work provides an
estimate of several important metrics used in SC-SM experiments to determine
the nature of ZBCPs, including the robustness, quantization, and end-to-end
correlation of the trivial ZBCPs. Therefore, in order to claim any evidence for
MZMs, one must establish the observed ZBCPs to have considerable statistical
significance well beyond what we find in this work to exist for the trivial
peaks.
