  Inspired by "quantum graphity" models for spacetime, a statistical model of graphs is proposed to explore possible realizations of emergent manifolds. Graphs with given numbers of vertices and edges are considered, governed by a very general Hamiltonian that merely favors graphs with near-constant valency and local rotational symmetry. The ratio of vertices to edges controls the dimensionality of the emergent manifold. The model is simulated numerically in the canonical ensemble for a given vertex to edge ratio, where it is found that the low-energy states are almost triangulations of two-dimensional manifolds. The resulting manifold shows topological "handles" and surface intersections in a higher embedding space, as well as non-trivial fractal dimension consistent with previous spectral analysis, and nonlocal links consistent with models of disordered locality. The transition to an emergent manifold is first order, and thus dependent on microscopic structure. Issues involved in interpreting nearly-fixed valency graphs as Feynman diagrams dual to a triangulated manifold as in matrix models are discussed. Another interesting phenomenon is that the entropy of the graphs are super-extensive, a fact known since Erd\H{o}s, which results in a transition temperature of zero in the limit of infinite system size: infinite manifolds are always disordered. Aside from a finite universe or diverging coupling constraints as possible solutions to this problem, long-range interactions between vertex defects also resolve the problem and restore a nonzero transition temperature, in a manner similar to that in low-dimensional condensed-matter systems. 
Statistical mechanics of graph models and their implications for   emergent spacetime manifolds