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

Abstract

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 $W_N$ 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.