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Velocity Polytopes of Periodic Graphs and a No-Go Theorem for Digital Physics

A periodic graph in dimension $d$ is a directed graph with a free action of $\Z^d$ with only finitely many orbits. It can conveniently be represented in terms of an associated finite graph with weights in $\Z^d$, corresponding to a $\Z^d$-bundle with connection. Here we use the weight sums along cycles in this associated graph to construct a certain polytope in $\R^d$, which we regard as a geometrical invariant associated to the periodic graph. It is the unit ball of a norm on $\R^d$ describing the large-scale geometry of the graph. It has a physical interpretation as the set of attainable velocities of a particle on the graph which can hop along one edge per timestep. Since a polytope necessarily has distinguished directions, there is no periodic graph for which this velocity set is isotropic. In the context of classical physics, this can be viewed as a no-go theorem for the emergence of an isotropic space from a discrete structure.