Concentration and central limit theorem for the averaging process on ${\mathbb{Z}}^d$

Abstract

In the averaging process on a graph $G=(V,E)$, a random mass distribution $\eta$ on $V$ is repeatedly updated via transformations of the form ${\eta }_v,{\eta }_w\mapsto ({\eta }_v+{\eta }_w)/2$, with updates made according to independent Poisson clocks associated to the edge set $E$. We study the averaging process when $G$ is the integer lattice ${\mathbb{Z}}^d$. We prove that the process has tight asymptotic concentration around its mean in the ${\ell }^1$ and ${\ell }^2$ norms and use this to prove a central limit theorem. Previous work by Nagahata and Yoshida implies the central limit theorem when $d\ge 3$. Our results extend this to hold for all $d\ge 1$, and our techniques are likely applicable to other processes for which previously only the $d\ge 3$ case was tractable.