  We consider i.i.d. last-passage percolation on $\mathbb{Z}^2$ with weights
having distribution $F$ and time-constant $g_F$. We provide an explicit
condition on the large deviation rate function for independent sums of $F$ that
determines when some adjacent Busemann function increments are negatively
correlated. As an example, we prove that $\operatorname{Bernoulli}(p)$ weights
for $p > p^* \approx 0.6504$ satisfy this condition. We prove this condition by
establishing a direct relationship between the negative correlations of
adjacent Busemann increments and the dominance of the time-constant $g_F$ by
the function describing the time-constant of last-passage percolation with
exponential or geometric weights.
