  The discrete Klein-Gordon equation on a two-dimensional square lattice
satisfies an $\ell^1 \mapsto \ell^\infty$ dispersive bound with polynomial
decay rate $|t|^{-3/4}$. We determine the shape of the light cone for any
choice of the mass parameter and relative propagation speeds along the two
coordinate axes. Fundamental solutions experience the least dispersion along
four caustic lines interior to the light cone rather than along its boundary,
and decay exponentially to arbitrary order outside the light cone. The overall
geometry of the propagation pattern and its associated dispersive bounds are
independent of the particular choice of parameters. In particular there is no
bifurcation of the number or type of caustics that are present.
  The discrete Klein-Gordon equation is a classical analogue of the quantum
harmonic lattice. In the quantum setting, commutators of time-shifted
observables experience the same decay rates as the corresponding Klein-Gordon
solutions, which depend in turn on the relative location of the observables'
support sets.
