  This paper is devoted to an interacting particle system that provides
probabilistic interpretation of the wave equation on graphs. A Feynman-Kac-type
formula is established, connecting the expectation of the process with the wave
equation on graphs. Non-asymptotic $L^2$ estimates are presented. It is then
shown that the high-density hydrodynamic limit of the system is given by the
wave equation in Euclidean space. The sharpness of scaling limit result is
demonstrated by a phase transition phenomenon.
