Random Assignment Problems on 2d Manifolds

We consider the assignment problem between two sets of $N$ random points on a smooth, two-dimensional manifold $\Omega$ unit area. It is known that average cost scales as $E_{\Omega}(N)\sim\frac{1}{2\pi}\ln N$ with correction at most order $\sqrt{\ln N\ln\ln N}$. In this paper, we show that, within linearization approximation field-theoretical formulation problem, first $\Omega$-dependent constant term, and can be exactly computed from spectrum Laplace--Beltrami operator $\Omega$. perform explicit calculation for various families surfaces, compare our predictions extensive numerics.