Type: Article
Publication Date: 2021-05-01
Citations: 5
DOI: https://doi.org/10.1007/s10955-021-02768-4
Abstract We consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>Ω</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mo>ln</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math> with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msqrt><mml:mrow><mml:mo>ln</mml:mo><mml:mi>N</mml:mi><mml:mo>ln</mml:mo><mml:mo>ln</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:msqrt></mml:math> . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.