${2d}$流形上的随机指派问题
Random assignment problems on ${2d}$ manifolds
摘要：
研究单位面积光滑二维流形$Ω$上两组$N$随机点之间的指派问题。已知平均成本缩放为具有至多为$\sqrt{\ln N\ln\ln N}$阶的校正的$E_Ω(N)\sim\frac{1}{2π}\ln N$。在本文中，我们表明，在线性化近似的领域理论制定的问题，第一个$Ω$相关的修正是在常数项，并可以准确地计算从频谱的拉普拉斯-贝尔特拉米算子$Ω$。我们执行明确的计算这个常数的各种家庭的表面，并比较我们的预测与广泛的数值。
英文摘要：
We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $Ω$ of unit area. It is known that the average cost scales as $E_Ω(N)\sim\frac{1}{2π}\ln N$ with a correction that is at most of order $\sqrt{\ln N\ln\ln N}$. In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $Ω$-dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace--Beltrami operator on $Ω$. We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.
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。
