Geometrogenesis under quantum graphity: Problems with the ripening universe

Quantum graphity (QG) is a model of emergent geometry in which space is represented by a dynamical graph. The graph evolves under the action of a Hamiltonian from a high-energy pregeometric state to a low-energy state in which geometry emerges as a coarse-grained effective property of space. Here we show the results of numerical modeling of the evolution of the QG Hamiltonian, a process we term ripening by analogy with crystallographic growth. We find that the model as originally presented favors a graph composed of small disjoint subgraphs. Such a disconnected space is a poor representation of our Universe. A new term is introduced to the original QG Hamiltonian, which we call the hypervalence term. It is shown that the inclusion of a hypervalence term causes a connected latticelike graph to be favored over small isolated subgraphs.