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(nd/(d+1)). Using partitioning
techniques, sub-additivity and concentration of measure
arguments, we show first that, after normalization by n2/3, the size of two-dimensional
square packings tends in probability toward a genuine limit γ. Straightforward concentration
arguments show that large fluctuations of order n2/3 should have probability vanishing
exponentially fast with n2/3. Even though γ remains unknown, using a change of measure
argument we show that this upper bound on tail probabilities is qualitatively correct.