AI 中文总结
研究图式两选择分配中端点局部规则的负载差距下界，通过扩散影响分析证明环上差距至少为常数乘以$\min\{\sqrt n,t^{1/4}\}$，并推广到环面，区分局部与全局策略。
AI 中文摘要
在图式两选择分配中，每个到达的球被分配到随机边的一个端点。我们研究决策是两端负载的单调函数的规则，允许依赖于边的阈值和新的随机化。这样的规则具有精确的单位差异耦合：在初始状态中增加一个球会在之后的每个时刻产生一个标记的差异。我们通过条件期望投影在标记边空间上表示标记，并获得扩散位移界。一个传输体积不等式将影响的缓慢传播转化为负载差距的下界。在具有$n$个顶点的环上，从任意初始分布出发，在任意物理时间$t\ge 1/n$，期望差距至少为常数乘以$\min\{\sqrt n,t^{1/4}\}$，且差距超过此尺度的概率至少为$1/8$。在恰好$k\ge1$次分配后，相应的尺度为$\min\{\sqrt n,(k/n)^{1/4}\}$。未使用平稳性、对称性、递归或矩假设。该一般不等式还给出了$L\times K$离散环面$C_L\square C_K$上阶为$\sqrt{L/K+\log K}$的下界。这些结果将端点局部规则与在环上实现多对数差距的全局信息策略区分开来。
英文摘要
In graphical two-choice allocation, each arriving ball is assigned to one endpoint of a random edge. We study rules whose decision is a monotone function of the two endpoint loads, allowing edge-dependent thresholds and fresh randomization. Such a rule has an exact unit-discrepancy coupling: adding one ball to the initial state produces one tagged discrepancy at every later time. We represent the tag by conditional-expectation projections on the marked edge space and obtain diffusive displacement bounds. A transport-volume inequality then converts slow propagation of influence into lower bounds for the load gap. On the cycle with $n$ vertices, from every initial distribution and at every physical time $t\ge 1/n$, the expected gap is at least a constant times $\min\{\sqrt n,t^{1/4}\}$, and the gap exceeds this scale with probability at least $1/8$. After exactly $k\ge1$ allocations, the corresponding scale is $\min\{\sqrt n,(k/n)^{1/4}\}$. No stationarity, symmetry, recurrence, or moment assumption is used. A smoothed threshold rule in the same class has expected gap $O(\sqrt n\log n)$ up to any fixed polynomial time horizon, so the saturated cycle bound is sharp within the class up to a logarithmic factor. The general inequality also yields a lower bound of order $\sqrt{L/K}$ on the $L\times K$ rectangular torus $C_L\square C_K$; combined with a strategy-independent logarithmic bound, this gives order $\sqrt{L/K}+\log(LK)$. These results separate endpoint-local rules from global-information strategies that achieve polylogarithmic gaps on cycles. All numbered results are verified in Lean.
Comments28 pages, 2 figures. All numbered results are formalized in Lean 4: https://github.com/obinnaokechukwu/cycle-gap-lowerbound-lean
