# Probability Distributions – What is the expected minimum total distance between two partitions identical and independent of distributed points?

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:

1. Caracciolo, S. & Sicuro, G. (2015). Square stochastic Euclidean
two-part adjustment
problem
,
Physical overview letters115 (23), 230601.
2. Holroyd, A.E., R. Pemantle, Y. Peres and O. Schramm (2009).
Poisson
suitable
,
in the Annales de l'Institut Henri Poincaré, Probabilités et
Statistiques
(Vol. 45, No. 1, pages 266-287). Institute Henri
Poincaré.
3. Boniolo, E., Caracciolo, S. & Sportiello, A. (2014). correlation
Function for the Euclidean grid Poisson matching on one line and on one
circle
, Journal of
Statistical Mechanics: Theory and Experiment
2014 (11), P11023.

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.