一维随机指派问题最优费用的反常标度
Anomalous scaling of the optimal cost in the one-dimensional random assignment problem
摘要：
我们考虑两组$N$随机点之间的直线上的随机欧几里德分配问题，这两组$N$随机点独立地由相同的概率密度函数$\varrho$生成。匹配的成本应该取决于匹配对的欧几里德距离的幂$p>1$。我们讨论了$N\gg 1$的平均最优成本的积分表达式，推广了以前的结果$p=2$。我们还研究了由于概率密度函数的消失而可能导致的表达式的发散。所提供的正则化方法允许我们在发散情况下恢复成本的适当比例律，以及可能的一些相关系数。本文还分析了$\varrho$的支持是不连通区间的可能性。我们将建议的程序，我们比较我们的预测与数值模拟的结果。
英文摘要：
We consider the random Euclidean assignment problem on the line between two sets of $N$ random points, independently generated with the same probability density function $\varrho$. The cost of the matching is supposed to be dependent on a power $p>1$ of the Euclidean distance of the matched pairs. We discuss an integral expression for the average optimal cost for $N\gg 1$ that generalizes a previous result obtained for $p=2$. We also study the possible divergence of the given expression due to the vanishing of the probability density function. The provided regularization recipe allows us to recover the proper scaling law for the cost in the divergent cases, and possibly some of the involved coefficients. The possibility that the support of $\varrho$ is a disconnected interval is also analysed. We exemplify the proposed procedure and we compare our predictions with the results of numerical simulations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
