Random assignment problems on $2d$ manifolds

Abstract

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.

34 pages, 7 figures

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Cited by in corpus (2)

• Asymptotic quantization on Riemannian manifolds via covering growth estimates
• Asymptotics of the quantization problem on metric measure spaces