Intrinsic correlation in planar Poisson line processes

The polygons arising from a planar Poisson line process have an exponential distribution of their side lengths and are known to become more regular as their area, perimeter or number of sides increases. Local regions with higher line density have smaller polygon side lengths and conversely. Numerical analysis of computer generated Poisson line processes shows that when pairs of adjacent polygon sides ( x , y ) are sorted such that x ≤ y , they are correlated with  ρ ≈ 0.616 as compared to  ρ = 1 / 5 ≈ 0.447 for independent sorted exponential ( x , y ) pairs. This correlation is consistent with the observed regularity of polygons in realizations of planar Poisson line processes.