the ABS is pinned for Zeeman energies $\alpha/L\lesssim\Delta_Z\lesssim E_{so}$; for the parameters of Fig. \[fig2\], these energy scales correspond to $\alpha/L\approx1.8\Delta_0$ and $E_{so}\approx2\Delta_Z^c$. Additionally, we show in Fig. \[fig2\](c) that a pinned ABS can also be observed when the chemical potential is detuned away from $\mu=0$. We plot the probability density of the ABS within the nanowire in Fig. \[wf\], showing that most of the weight within the nanowire is concentrated in the quantum dot; however, we also find that the wave function of the ABS is roughly spread equally between the nanowire and the superconducting layer.

To check the analytical result that the presence of a low-energy ABS is sensitive only to parameters of the normal dot region, we can test the robustness of the ABS to variations in both the thickness $d$ of the superconducting layer as well as the strength of the proximity coupling $t$ [@disorder]. It was shown in Refs. [@Reeg:2017_3; @Reeg:2018] that all properties of the region of nanowire coupled to the superconductor are highly dependent on both the thickness $d$ and the tunneling strength $t$. In particular, the effective chemical potential shift, and therefore the field strength at which a topological phase transition occurs, 