We apply an invariance principle due to De Masi, Ferrari, Goldstein and Wick to the edge process for critical reversible nearest-particle systems. Their result also gives an upper bound for the diffusion constant that we compute explicitly. A comparison between the movement of the edge, when the other particles are frozen, and a random walk allows us to find a lower bound for the diffusion constant. This shows that the right renormalization for the edge to converge to a nondegenerate Brownian motion is the usual one. Note that analogous results for nearest-particle systems are only known for the contact process in the supercritical case.
Rinaldo Schinazi. "Brownian Fluctuations of the Edge for Critical Reversible Nearest-Particle Systems." Ann. Probab. 20 (1) 194 - 205, January, 1992. https://doi.org/10.1214/aop/1176989924