Sharp Bounds for Dynamic Averaging on Cycles

We study a dynamic averaging process on the cycle $C_n$ with bounded, time-varying load arrivals. At each discrete time $t$, an edge is chosen uniformly at random, a load $0\le w_t\le 1$ is introduced, and the total load of its two endpoints together with $w_t$ is divided equally between them. Starting from the flat configuration, we prove that the expected gap between the largest and smallest loads is $O(\sqrt{n})$, uniformly in time and over all such arrival sequences. Building on the lower-bound argument of Alistarh, Nadiradze, and Sabour for the expected square of the gap, we further show that whenever the loads are uniformly bounded away from $0$, the expected gap is $\mathrm{\Omega }(\sqrt{n})$ for all sufficiently large times. In particular, this confirms their conjecture that the expected gap is of order $\sqrt{n}$.