Random Assignment Problems on 2d Manifolds
Résumé
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)] ∼ 1/2π 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.
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