We connect the zero energy states of bipartite tight-binding models with bond disorder and vacancy disorder, as well as collective zero-energy Majorana excitations of networks (not necessarily bipartite) of coupled localized Majorana modes to the random geometry of regions of the underlying disordered lattice which host the monomers of any maximum-density dimer packing of the lattice. This allows us to place upper bounds on the localization length of the zero-energy Green function in both cases, and use these upper bounds to draw conclusions about heat transport in the Majorana networks or electrical transport in binary alloys described by such bipartite random hopping models. For the Majorana contribution to the heat conductance in the mixed triangular vortex lattice state of topological p+ip superconductors, we argue that our results imply a strong violation of self-averaging if the positional disorder of the vortex lattice is strong, and the vacancy-disorder is weak.

[collaborators: Sounak Biswas, Mursalin Islam, and Ritesh Bhola]