  The classic combinatorial construct of {\em maximum matchings} probes the
random geometry of regions with local sublattice imbalance in a site-diluted
bipartite lattice. We demonstrate that these regions, which host the monomers
of any maximum matching of the lattice, control the localization properties of
a zero-energy quantum particle hopping on this lattice. The structure theory of
Dulmage and Mendelsohn provides us a way of identifying a complete and
non-overlapping set of such regions. This motivates our large-scale
computational study of the Dulmage-Mendelsohn decomposition of site-diluted
bipartite lattices in two and three dimensions. Our computations uncover an
interesting universality class of percolation associated with the end-to-end
connectivity of such monomer-carrying regions with local sublattice imbalance,
which we dub {\em Dulmage-Mendelsohn percolation}. Our results imply the
existence of a monomer percolation transition in the classical statistical
mechanics of the associated maximally-packed dimer model and the existence of a
phase with area-law entanglement entropy of arbitrary many-body eigenstates of
the corresponding quantum dimer model. They also have striking implications for
the nature of collective zero-energy Majorana fermion excitations of bipartite
networks of Majorana modes localized on sites of diluted lattices, for the
character of topologically-protected zero-energy wavefunctions of the bipartite
random hopping problem on such lattices, and thence for the corresponding
quantum percolation problem, and for the nature of low-energy magnetic
excitations in bipartite quantum antiferromagnets diluted by a small density of
nonmagnetic impurities.
