  Quantum graphity is a background independent model for emergent locality,
spatial geometry and matter. The states of the system correspond to dynamical
graphs on N vertices. At high energy, the graph describing the system is highly
connected and the physics is invariant under the full symmetric group acting on
the vertices. We present evidence that the model also has a low-energy phase in
which the graph describing the system breaks permutation symmetry and appears
to be ordered, low-dimensional and local. Consideration of the free energy
associated with the dominant terms in the dynamics shows that this low-energy
state is thermodynamically stable under local perturbations. The model can also
give rise to an emergent U(1) gauge theory in the ground state by the
string-net condensation mechanism of Levin and Wen. We also reformulate the
model in graph-theoretic terms and compare its dynamics to some common graph
processes.
