Random Assignment Problems on 2d Manifolds

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 ) ∼ 1 / 2 π ln N with a correction that is at most of order 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.