论文标题
$ {2D} $歧管上的随机分配问题
Random assignment problems on ${2d}$ manifolds
论文作者
论文摘要
我们考虑两组$ n $随机点之间的分配问题，在平滑的二维歧管$ω$的单位区域之间。众所周知，平均成本量表为$e_Ω（n）\ sim \ frac {1} {2π} \ ln n $，其校正大多是$ \ sqrt {\ sqrt {\ ln n n \ ln \ ln \ ln n} $。在本文中，我们表明，在问题的字段理论公式的线性化近似中，第一个$ω$依赖性校正是在恒定项上，并且可以从$ω$上的laplace-beltrami操作员的光谱中精确计算。我们对各个表面族的族进行了明确的计算，并将我们的预测与广泛的数字进行比较。
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.
