  We investigate the relationship between the sizes of the sum and difference
sets attached to a subset of {0,1,...,N}, chosen randomly according to a
binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the
random subset is almost surely difference dominated, as N --> oo, for any
choice of p(N) tending to zero, thus confirming a conjecture of Martin and
O'Bryant. The proofs use recent strong concentration results.
  Furthermore, we exhibit a threshold phenomenon regarding the ratio of the
size of the difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all
sums and differences in the random subset are almost surely distinct, and in
particular the difference set is almost surely about twice as large as the
sumset. If N^{-1/2} = o(p(N)) then both the sum and difference sets almost
surely have size (2N+1) - O(p(N)^{-2}), and so the ratio in question is almost
surely very close to one. If p(N) = c N^{-1/2} then as c increases from zero to
infinity (i.e., as the threshold is crossed), the same ratio almost surely
decreases continuously from two to one according to an explicitly given
function of c.
  We also extend our results to the comparison of the generalized difference
sets attached to an arbitrary pair of binary linear forms. For certain pairs of
forms f and g, we show that there in fact exists a sharp threshold at c_{f,g}
N^{-1/2}, for some computable constant c_{f,g}, such that one form almost
surely dominates below the threshold, and the other almost surely above it.
  The heart of our approach involves using different tools to obtain strong
concentration of the sizes of the sum and difference sets about their mean
values, for various ranges of the parameter p.
