  Inspired by the prospect of having discretized spaces emerge from random
graphs, we construct a collection of simple and explicit exponential random
graph models that enjoy, in an appropriate parameter regime, a roughly constant
vertex degree and form very large numbers of simple polygons (triangles or
squares). The models avoid the collapse phenomena that plague naive graph
Hamiltonians based on triangle or square counts. More than that, statistically
significant numbers of other geometric primitives (small pieces of regular
lattices, cubes) emerge in our ensemble, even though they are not in any way
explicitly pre-programmed into the formulation of the graph Hamiltonian, which
only depends on properties of paths of length 2. While much of our motivation
comes from hopes to construct a graph-based theory of random geometry
(Euclidean quantum gravity), our presentation is completely self-contained
within the context of exponential random graph theory, and the range of
potential applications is considerably more broad.
