Negative correlation of adjacent Busemann increments

TL;DR

This paper identifies conditions under which adjacent Busemann increments in last-passage percolation are negatively correlated, linking this to large deviation principles and specific weight distributions like Bernoulli.

Contribution

It provides an explicit condition on the large deviation rate function that determines negative correlation of Busemann increments in last-passage percolation models.

Findings

Negative correlation of Busemann increments for certain weight distributions.

Explicit condition involving large deviation rate functions.

Bernoulli weights with p > 0.6504 satisfy the negative correlation condition.

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 $\operatorname{Bernoulli}(p)$ 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.