Concentration and central limit theorem for the averaging process on Zᵈ

In the averaging process on a graph $G = (V, E)$, a random mass distribution η on V is repeatedly updated via transformations of the form ηᵥ, ηw ↦ (ηᵥ + ηw)/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 Zᵈ. 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.