Brownian motion as limit of the interchange process

TL;DR

This paper proves that the rescaled trajectories of particles in the interchange process on path graphs converge to Brownian motion, establishing a law of large numbers for these particle trajectories.

Contribution

It demonstrates a weak convergence of empirical measures of particle trajectories to Brownian motion, providing a new law of large numbers result for the interchange process.

Findings

Empirical measure converges to Brownian motion

Rescaled particle trajectories follow a law of large numbers

Convergence holds on path graphs

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.