We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $\Omega$ of unit area. It is known that the average cost scales as $E_{ \Omega}(N)\sim\frac{ 1}{ 2\pi}\ln N$ 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 $\Omega$-dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace--Beltrami operator on $\Omega$. We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.
Categories: math-ph cond-mat.dis-nn math.MP math.PR