AI 中文总结
本文研究随机分配中密码符号数的最优选择，发现当重复模式信息被揭示时，若归一化轮廓超过临界值0.507834，最优支撑分数固定为1/2，并揭示系综非等价现象。
AI 中文摘要
一个密码应该使用多少个不同的符号？如果其长度固定为$n$，且观察者仅知道哪些符号出现，则当$k/n\to1/(2\log2)$时，兼容密码的数量渐近地最大化。我们探究当除长度外，还揭示关于重复模式的聚合信息时，情况会如何变化。我们通过固定第二个加性轮廓$V_k=\sum_i v(J_i)$（在尺度$V_k/n\approx\rho$下）来对此建模。对于$v(j)=\log(j!)$，该轮廓记录了由重复引起的兼容词数量对数减少量；我们证明，一旦归一化轮廓$\rho$超过$0.507834\ldots$，极限最优分数便被固定在$1/2$。对于上述阈值以上最优支撑下的典型重数轮廓，$V_k$中的超出部分由消失比例的已用符号承担。我们将此解释为系综非等价的一种形式，并将该机制推广到其他轮廓和非均匀分配模型。
英文摘要
How many distinct symbols should a password use? If its length is fixed at $n$ and an observer learns only which symbols appear, the number of compatible passwords is maximized asymptotically when $k/n\to1/(2\log2)$. We ask what changes when, in addition to the length, aggregate information about the repetition pattern is revealed. We model this by fixing a second additive profile $V_k=\sum_i v(J_i)$ at scale $V_k/n\approxρ$. For $v(j)=\log(j!)$, the profile records the reduction in the logarithm of the number of compatible words caused by repetitions; we show that once the normalized profile $ρ$ exceeds $0.507834\ldots$, the limiting optimal fraction is pinned at $1/2$. For a typical multiplicity profile at the optimal support above this threshold, the excess in $V_k$ is carried by a vanishing fraction of used symbols. We interpret this as a form of non-equivalence of ensembles and extend the mechanism to other profiles and non-uniform allocation models.
Comments28 pages, 3 figures
