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arXiv · 2008.01462

Random assignment problems on ${2d}$ manifolds

Dario Benedetto·Emanuele Caglioti·Sergio Caracciolo·Matteo D'Achille·Gabriele Sicuro·Andrea Sportiello
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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.

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Dario Benedetto, Emanuele Caglioti, Sergio Caracciolo, Matteo D'Achille, Gabriele Sicuro, Andrea Sportiello. 2020-08-04. Random assignment problems on ${2d}$ manifolds. https://doi.org/10.1007/s10955-021-02768-4

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