  Hardy conjectured that the error term arising from approximating the number
of lattice points lying in a radius-$R$ disc by its area is $O(R^{1/2+o(1)})$.
One source of support for this conjecture is a folklore heuristic that uses
i.i.d. random variables to model the lattice points lying near the boundary and
square-root cancellation of sums of these random variables. We examine this
heuristic by studying how these lattice points interact with one another and
prove that their autocorrelation is determined in terms of a random model.
Additionally, it is shown that lattice points near the boundary which are "well
separated" behave independently. We also formulate a conjecture concerning the
distribution of pairs of these lattice points.
