AI 中文总结
研究教皇秘密会议投票的随机模型，核心方法是设定投票概率与票数幂次成正比。贡献为揭示\(\alpha = 1\)时吸收时间的转变，证明\(\alpha>1\)时吸收时间降为\(\textit{loglog n}\)量级及获胜者身份的相变，表明强化投票能快速达成共识。
AI 中文摘要
我们引入了一个关于教皇秘密会议的随机模型，其中\(n\)位红衣主教相互反复投票，直到有一位红衣主教获得所有选票。在每一轮中，一位红衣主教投票给给定候选人的概率与该候选人在上一轮的票数的\(\alpha\)次幂成正比。当\(\alpha = 1\)时，该模型简化为赖特 - 费希尔模型，并且与金曼的\(n\) - 合并过程对偶。我们揭示了在\(\alpha = 1\)时吸收时间\(\mathcal{T}\)的急剧转变。已知当\(\alpha = 1\)时，\(\mathcal{T}\)通常是\(n\)的量级。我们证明对于\(\alpha>1\)，它降至\(\textit{loglog n}\)量级。相反，对于\(\alpha<1\)，\(\mathcal{T}\)通常至少是\(\exp(\Omega(n))\)。我们还证明了当\(\alpha>1\)时获胜者身份的急剧相变。对于每个正整数\(k\)，如果\(2^{1/k}<\alpha<2^{1/(k - 1)}\)（其中我们记\(2^{1/0}=+\infty\)），随着\(n\to\infty\)，最终获胜者以概率趋于\(1\)是第\(k\)轮后的唯一领导者。这些结果表明，即使对于大型选民群体，强化投票过程也能非常迅速地达成共识。
英文摘要
We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $α$-th power of that candidate's vote count in the preceding round. For $α=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's n-coalescent. We reveal a sharp transition in the absorption time $\mathcal{T}$ at $α=1$. It was known that when $α=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $α>1$, it drops to order $\textit{loglog n.}$ In contrast, for $α<1$, $\mathcal{T}$ is typically at least $\exp(Ω(n))$. We also prove a sharp phase transition in the identity of the winner when $α>1$. For every positive integer $k$, if $2^{1/k}<α<2^{1/(k-1)}$ (where we write $2^{1/0} = +\infty$), with probability tending to 1 as $n\to\infty$, the eventual winner is the unique leader after round $k$. These results show that reinforced voting processes reach consensus remarkably quickly even for large electorates.
