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The classic combinatorial construct of {\em maximum matchings} probes therandom geometry of regions with local sublattice imbalance in a site-dilutedbipartite lattice. We demonstrate that these regions, which host the monomersof any maximum matching of the lattice, control the localization properties ofa zero-energy quantum particle hopping on this lattice. The structure theory ofDulmage and Mendelsohn provides us a way of identifying a complete andnon-overlapping set of such regions. This motivates our large-scalecomputational study of the Dulmage-Mendelsohn decomposition of site-dilutedbipartite lattices in two and three dimensions. Our computations uncover aninteresting universality class of percolation associated with the end-to-endconnectivity of such monomer-carrying regions with local sublattice imbalance,which we dub {\em Dulmage-Mendelsohn percolation}. Our results imply theexistence of a monomer percolation transition in the classical statisticalmechanics of the associated maximally-packed dimer model and the existence of aphase with area-law entanglement entropy of arbitrary many-body eigenstates ofthe corresponding quantum dimer model. They also have striking implications forthe nature of collective zero-energy Majorana fermion excitations of bipartitenetworks of Majorana modes localized on sites of diluted lattices, for thecharacter of topologically-protected zero-energy wavefunctions of the bipartiterandom hopping problem on such lattices, and thence for the correspondingquantum percolation problem, and for the nature of low-energy magneticexcitations in bipartite quantum antiferromagnets diluted by a small density ofnonmagnetic impurities.

Dulmage-Mendelsohn percolation: Geometry of maximally-packed dimer models and topologically-protected zero modes on site-diluted bipartite lattices