环上动态平均过程的尖锐界
Dynamic Averaging on Regular Graphs
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中文总结 AI 辅助
研究环上动态平均过程，证明最大最小负载期望差距为Θ(√n)，解决了Alistarh等人的猜想。
中文摘要 AI 辅助
我们研究环\\(C_n\\)上的动态平均过程。在每个离散时刻，均匀随机选择一条边，引入一个单位的负载，并将两个端点的负载替换为加入新单位后的共同平均值。从零配置开始，我们证明最大负载与最小负载之间的期望差距一致地为\\(O(\sqrt n)\\)。基于Alistarh、Nadiradze和Sabour关于期望平方差距的下界论证，我们进一步证明长期来看期望差距为\\(\Omega(\sqrt n)\\)。这证实了他们的猜想，即期望差距的量级为\\(\sqrt n\\)。
英文摘要
We study a dynamic averaging process on finite regular graphs 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 obtain a pairwise concentration bound governed by the effective resistance between vertices and use generic chaining to derive a general upper bound on the expected gap between the largest and smallest loads. As a consequence, we show that every $d$-regular graph has expected gap $O_d(\sqrt n)$, uniformly in time and over all deterministic arrival sequences. Applying our general bound to the discrete two-dimensional torus yields the sharp $O(\log n)$ upper bound, improving the best previously known bound. For the cycle, whenever the arriving loads are bounded away from zero, we prove that the expected gap is $Ω(\sqrt n)$ for all sufficiently large times. Together with our upper bound, this confirms the conjecture of Alistarh, Nadiradze, and Sabour that the expected gap on the cycle is of order $\sqrt n$.
