Random Assignment Problems on 2d Manifolds
Abstract
We consider the assignment problem between two sets of N random points on a smooth, twodimensional manifold Ω of unit area. It is known that the average cost scales as E_{Ω}(N ) ∼1 /2 π lnN with a correction that is at most of order √{lnN lnlnN }. In this paper, we show that, within the linearization approximation of the fieldtheoretical formulation of the problem, the first Ω dependent correction is on the constant term, and can be exactly computed from the spectrum of the LaplaceBeltrami operator on Ω . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.
 Publication:

Journal of Statistical Physics
 Pub Date:
 May 2021
 DOI:
 10.1007/s10955021027684
 arXiv:
 arXiv:2008.01462
 Bibcode:
 2021JSP...183...34B
 Keywords:

 matching;
 assignment;
 random optimization problems;
 optimal transportation;
 finitesize corrections;
 disorder;
 Mathematical Physics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Mathematics  Probability
 EPrint:
 34 pages, 7 figures