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. Physical review letters, 115(23), 230601.
2. Holroyd, A. E., Pemantle, R., Peres, Y., & Schramm, O. (2009). Poisson matching. 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. 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?