  We verify signatures of antiferromagnetic (AF) correlations in the double
occupancy D [Gorelik et al., PRL 105, 065301 (2010)] and study their
dimensional dependence using direct quantum Monte Carlo in dimensions d=2,3 and
Bethe Ansatz in d=1. We find quantitative agreement with dynamical mean-field
theory (DMFT) in the cubic case and qualitative agreement down to d=1. As a
function of entropy s=S/(N k_B), D is nearly universal with respect to d; the
minimum in D(s) approaches s=log(2) at strong coupling, as predicted by DMFT.
Long-range order appears hardly relevant for the current search of AF
signatures in cold fermions. Thus, experimentalists need not achieve s<log(2)/2
and should consider lower dimensions, for which the AF effects are larger.
