Suppose a square $[0,1]times [0,1]$ in which $ N $ vehicles $ V_i $ and $ N $ equestrian $ R_i $ are distributed equally and independently (eg uniform distribution), a two-part adjustment (or a permutation), $ pi (i) $) is carried out between the vehicles and drivers with the aim of keeping the total distance

$$ Z = min _ { pi} sum_1 ^ N sqrt { Vert V _ { pi (i)} – R_i Vert ^ 2} $$

is minimized.

Because the locations of vehicles and drivers are therefore randomly distributed $ Z $ is a random variable. The expectation of $ Z $ is therefore of interest. The question is how to derive this $ E (Z) $,

I found some related papers, such as:

- Caracciolo, S. & Sicuro, G. (2015). Square stochastic Euclidean

two-part adjustment

problem,

*Physical overview letters*115 (23), 230601. - Holroyd, A.E., R. Pemantle, Y. Peres and O. Schramm (2009).

Poisson

suitable,

in the*Annales de l'Institut Henri Poincaré, Probabilités et*(Vol. 45, No. 1, pages 266-287). Institute Henri

Statistiques

Poincaré. - Boniolo, E., Caracciolo, S. & Sportiello, A. (2014). correlation

Function for the Euclidean grid Poisson matching on one line and on one

circle,*Journal of*2014 (11), P11023.

Statistical Mechanics: Theory and Experiment

I try to read them to find out how to do that, but their derivation has everything in part to do with physics and statistical mechanics, which makes it hard for me to understand, but I fail.

I was wondering if there is a version that does not know any physics just operations research to solve this problem.