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Random matrix theory for the robustness, quantization, and end-to-end correlation of zero-bias conductance peaks in a class D ensemble

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.