Dulmage-Mendelsohn percolation

We study the random geometry of maximum matchings of diluted bipartite lattices in two and three dimensions. Our study brings into play a graph-theoretical tool, the Dulmage-Mendelsohn decomposition of a bipartite graph. Using this, we define two classes of non-overlapping connected regions, R-type and P-type regions, that together cover the diluted lattice. We demonstrate that the size of these regions controls monomer and dimer correlations in the statistical mechanics of the corresponding maximally-packed dimer models. Additionally, we argue that R-type regions host topologically-protected zero modes in the quantum mechanics of a particle hopping on such random lattices. These R-type regions also host emergent Majorana fermion excitations of the corresponding bipartite Majorana networks. In two dimensions, we show  that the random geometry of R-type regions exhibits universal scaling behaviour controlled by a diverging localization length in the limit of vanishing dilution n -> 0. In three dimensions, we demonstrate that these R-type regions display critical scaling behaviour in the vicinity of a percolation transition at a nonzero dilution threshold n_c. In this case, we also establish the presence of a second transition at a lower dilution threshold n_l, corresponding to a spontaneous breaking of sublattice symmetry within the delocalized phase. [Based on work done @ TIFR by Ritesh Bhola, Sounak Biswas, & Md. Mursalin Islam, made possible by the computational resources of DTP-TIFR.]