Concentration and central limit theorem for the averaging process on Z^d

In the averaging process on a graph G = (V, E), a random mass distribution on V is repeatedly updated via transformations of the form ₕ, ₖ (ₕ + ₖ) /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 Z^d. We prove that the process has tight asymptotic concentration around its mean in the ^1 and ^2 norms and use this to prove a central limit theorem. Previous work by Nagahata and Yoshida implies the central limit theorem when d 3. Our results extend this to hold for all d 1, and our techniques are likely applicable to other processes for which previously only the d 3 case was tractable.