On a Square Packing Problem

Abstract

An instance of the square packing problem consists of n squares with independently, uniformly distributed side-lengths and independently, uniformly distributed locations on the unit d-dimensional torus. A packing is a maximum family of pairwise disjoint squares. The one-dimensional version of the problem is the classical random interval packing problem. This paper deals with the asymptotic behaviour of packings as n tends to infinity while d = 2. Coffman, Lueker, Spencer and Winkler recently proved that the average size of packing is Θ(n^d/(d+1)). Using partitioning techniques, sub-additivity and concentration of measure arguments, we show first that, after normalization by n^2/3, the size of two-dimensional square packings tends in probability toward a genuine limit γ. Straightforward concentration arguments show that large fluctuations of order n^2/3 should have probability vanishing exponentially fast with n^2/3. Even though γ remains unknown, using a change of measure argument we show that this upper bound on tail probabilities is qualitatively correct.