Long range voter models and dynamical fractional Brownian motion

We study the voter model on Z with long-range interactions, as proposed by Hammond and Sheffield. We show a spacetime rescaling converges to a fractional Gaussian free field, which can be viewed as a one-parameter family of fractional Brownian motions. As a consequence, we obtain that long-range voter models rescale to fractional Gaussian noise. The argument uses the Lindeberg swapping technique and heat kernel estimates for random walks with jump distributions in the domain of attraction of a stable law.