We study a dynamic averaging process on the cycle $C_n$. At each discrete time, an edge is chosen uniformly at random, one unit of load is introduced, and the two endpoint loads are replaced by their common average after the new unit has been added. Starting from the zero configuration, we prove that the expected gap between the largest and smallest loads is $O(\sqrt{n})$, uniformly in time. Building on the lower-bound argument of Alistarh, Nadiradze, and Sabour for the expected square of the gap, we further show that the expected gap is $\Omega(\sqrt{n})$ in the long run. This confirms their conjecture that the expected gap is of order $\sqrt{n}$.
Blogger's Review: This paper provides a new perspective on dynamic averaging processes through rigorous mathematical derivation, especially on the specific structure of cycle graphs, revealing potential patterns in load balancing, which has significant theoretical implications and practical applications.