Order of decay of the wasted space for a stochastic packing problem

A packing of a collection of rectangles contained in [0,1]2 is a disjoint subcollection; the wasted space is the measure of the area of the part of [0,1]2 not covered by the subcollection.A simple packing has the further restriction that each vertical line meets at most one rectangle of the packing. Given a collection of N independent uniformly distributed subrectangles of [0,1], we proved in a previous work that there exists a number K such that the wasted space WN in an optimal simple packing of these rectangles satisfies for all N EW_N \leq \frac{K}{\sqrt{N}} \exp K\sqrt{\log N}. We prove here that \frac{1}{K\sqrt{N}} \exp \frac{1}{K} \sqrt{N} \leq EW_N.