Classical and Quantum Correlated Percolation

Percolation is the study of connected structures in disordered networks. As edges are randomly and independently added to the network, clusters of neighboring edges grow in size until there ultimately exists a percolating cluster. The emergence of this percolating cluster exhibits properties of a continuous phase transition. The simplicity of uncorrelated percolation makes it the Ising model of connectivity-driven phase transitions. This thesis documents a quest to go beyond uncorrelated percolation and investigate transitions in correlated percolation models where there are constraints on the addition of edges (or vertices). Such constraints are inspired by glassy and jamming systems, for example. More specifically, we discuss several correlated percolation models, the k-core model on random graphs, and the spiral and counter-balance models in two-dimensions---all exhibiting discontinuous transitions yet with diverging correlation lengths---in an effort to identify the needed ingredients for such a novel transition. We also construct mixtures of these models to interpolate between a continuous transition and a discontinuous transition to search for tricriticality. Then, to capture both the local and global mechanical stability properties of disordered particle packings, we work towards building jamming graphs via the Henneberg construction. This construction is another example of a correlated percolation model since there are constraints on the addition (and removal) of edges. Jamming graphs provide for a more rigorous way to define the jamming transition for repulsive soft spheres as well as a starting point to understanding how the local and global mechanical properties interact and characterize the system. In addition, we study quantum transport properties along these disordered networks with correlations in the connectivity and investigate how such correlations affect, for example, the transition from insulator to metal as the percolating cluster emerges. Treating each occupied edge as a quantum scatterer and invoking the random phase approximation, we find for k=3-core networks on the Bethe lattice (a connected, loop-free graph) that the random first-order phase transition in the connectivity also drives the onset of quantum conduction giving rise to a new type of metal-insulator transition. Finally, we conduct level spacing analysis to go beyond the random phase approximation. This analysis reiterates our findings of a new random first-order phase transition in the world of disorder-driven metal-insulator transitions