Negative correlation of adjacent Busemann increments

Abstract

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 $\mathrm{Bernoulli}(p)$ weights for $p>p^{\ast }\approx 0.6504$ 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.

22 pages, 4 figures. Generalized the main theorem to pre-Busemann functions, fixed several typos, and added plots of the time-constant