Suppose a square $[0,1]\times [0,1]$ in which $N$ vehicles $V_i$ and $N$ riders $R_i$ are distributed identically and independently (say, uniform distribution), a bipartite matching (or a permutation, $\pi(i)$) is done between the vehicles and riders with the objective that the total distance 

$$Z=\min_{\pi}\sum_1^N \sqrt{\Vert V_{\pi(i)}-R_i\Vert^2}$$

is minimized.

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

I have found some related papers, such as

 1. Caracciolo, S., & Sicuro, G. (2015). [Quadratic stochastic Euclidean
    bipartite matching
    problem](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.115.230601).
    *Physical review letters*, 115(23), 230601.    
 2. Holroyd, A. E., Pemantle, R., Peres, Y., & Schramm, O. (2009).
    [Poisson
    matching](http://www.numdam.org/article/AIHPB_2009__45_1_266_0.pdf).
    In *Annales de l'Institut Henri Poincaré, Probabilités et
    Statistiques* (Vol. 45, No. 1, pp. 266-287). Institut Henri
    Poincaré. 
 3. Boniolo, E., Caracciolo, S., & Sportiello, A. (2014). [Correlation
    function for the Grid-Poisson Euclidean matching on a line and on a
    circle](https://arxiv.org/pdf/1403.1836.pdf). *Journal of
    Statistical Mechanics: Theory and Experiment*, 2014(11), P11023.

I am trying to read them to figure out how, but their derivation all has some part heavily related to physics and statistical mechanics, which makes me to struggle to understand it but I fail.

I was wondering if there is a version with a no-physics-just-operations-research flavor to solve this problem?