Brownian motion as limit of the interchange process

Abstract

We prove that the random empirical measure of appropriately rescaled particle trajectories of the interchange process on path graphs converges weakly to the deterministic measure of stationary Brownian motion on the unit interval. This is a law of large numbers type result for particle trajectories of the interchange process. After the completion of this manuscript we learned about a result of Durrett and Neuhauser that implies this result.