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.

💡 Research Summary

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The paper studies infinite directed graphs that are periodic with respect to the free action of the integer lattice ℤⁿ. Such a “periodic graph” Γ is defined by requiring that the ℤⁿ‑action be free and that there be only finitely many vertex‑orbits and edge‑orbits. The authors introduce a finite combinatorial model for Γ, called a displacement graph (G, δ). Here G is the quotient graph Γ/ℤⁿ (a finite directed multigraph) and δ assigns to each edge e∈E(G) an integer vector δ(e)∈ℤⁿ describing the lattice translation needed to go from the source to the target of any lift of e in the covering graph Γ. This construction is equivalent to a voltage (or gain) graph and can be viewed as a ℤⁿ‑principal bundle with a flat connection.

The central geometric object is built from the cycle space of the finite graph G. For any directed cycle c in G, the authors define its displacement Δ(c)=∑_{e∈c}δ(e)∈ℤⁿ. The set of all such displacements, taken over all cycles, is a subset of ℤⁿ. Its convex hull in ℝⁿ is a polytope P(Γ), which the authors call the “velocity polytope”. The polytope is the unit ball of a norm on ℝⁿ that captures the large‑scale geometry of the covering graph Γ.

To give this polytope a physical interpretation, the paper considers a classical point particle moving on Γ. Time is discrete; at each time step the particle traverses exactly one edge. A trajectory is a sequence of vertices (f₀,f₁,…). If the limit

u = lim_{k→∞} (f_k – f₀)/k

exists, u∈ℝⁿ is called a velocity vector of Γ. Theorem 19 proves that the set of all such velocity vectors coincides precisely with the polytope P(Γ). In other words, every point of P(Γ) can be realized as the asymptotic average displacement of some infinite walk, and conversely any asymptotic average displacement must lie inside P(Γ). This establishes a clean combinatorial description of the macroscopic motion possible on a periodic graph.

Because a polytope necessarily has flat faces and distinguished vertices, it cannot be a Euclidean ball. Theorem 28 (the “no‑go theorem”) shows that no periodic graph can have an isotropic velocity set; there will always be privileged directions where the maximal attainable speed differs from other directions. Consequently, a discrete structure of the type considered cannot give rise, in the classical limit, to an isotropic continuous space. This result directly challenges proposals in “digital physics” that aim to derive space‑time as an emergent continuum from a fundamentally discrete network (e.g., models based on quantum graphity, spin‑network analogues, or cellular automata on lattices).

The authors acknowledge that the velocity polytope coincides with objects previously studied under different names: Eon’s “cycle figures”, Kotani‑Sunada’s “polytope D”, and the “cycle figures” of voltage graph theory. Their contribution lies in emphasizing the physical interpretation as a velocity set and in deriving the isotropy obstruction.

Additional results include:

• Corollary 27, which uses the velocity polytope to give a criterion for the non‑existence of translation‑invariant graph morphisms between two periodic graphs.
• Proposition 22, showing that the polytope encodes the large‑scale geometry (asymptotic cone) of the covering graph.
• Explicit examples in dimensions two and three illustrating how the construction works for familiar lattices (square, hexagonal, cubic, etc.) and how the resulting polytopes are polygons or polyhedra with non‑spherical shape.

In summary, the paper provides a rigorous bridge between combinatorial graph theory, geometric group theory, and classical mechanics on discrete spaces. By proving that the attainable macroscopic velocities form a polytope rather than a sphere, it delivers a mathematically solid “no‑go” theorem for the emergence of isotropic space from periodic discrete structures, thereby setting clear limits on a broad class of digital‑physics models.