Two-dimensional Topology in Combinatorial Emergent Gravity

Combinatorial Emergent Gravity is an attempt towards non-perturbative description of quantum gravity where one explores ways in which geometry and gravitational dynamics emerge from non-geometric relational quantum microstates defined using combinatorial objects such as simple graphs. A microscopic dynamics is postulated to exist such that the topology and geometry of spacetime will emerge in the low-temperature phase of the system. We describe a model with a Hamiltonian that is quartic in the adjacency matrix of the graphs placed in a canonical ensemble of all simple graphs of a fixed vertex count. A simulated-annealing and parallel tempering algorithms are utilized to explore the ground states and finite-temperature phases of the system. The system is found to have two-dimensional geometric graphs as the ground states.