Harris' inequality

Harris' inequality states that two increasing events under an independent Bernoulli product measure are positively correlated.

Under a product measure on a finite Boolean lattice, an increasing event $E$ and a decreasing event $G$ obey the displayed inequality. The complement $G^c$ is an increasing event, so Harris' inequality gives $\mathbb{P}(E\cap G^c)\ge \mathbb{P}(E)\mathbb{P}(G^c)$. Subtract from $\mathbb{P}(E)$ to obtain the result. This implication includes degenerate Bernoulli distribution parameters $0$ and $1$.

An event in a partially ordered configuration space is increasing when changing coordinates upward cannot make a configuration leave the event.

For a sequence of nontrivial increasing events on finite product measures, write $p_x$ for the parameter at which the success probability is $x$. An additive sharp threshold means that $p_{1-\epsilon }-p_{\epsilon }\to 0$ for every fixed $0<\epsilon <1/2$. When the threshold parameter itself tends to zero, relative sharpness instead compares this window with $p_{1/2}$; additive sharpness alone is then a weaker assertion. The Friedgut-Kalai sharp threshold theorem provides an explicit additive bound for transitive increasing events.

An increasing event on a finite product of identically distributed coordinates is transitive if a group action preserving the event is transitive on the coordinate set. Its influences are all equal. Transitivity concerns coordinates, and does not assert transitivity on successful configurations.

An event $A$ in a partially ordered set is a decreasing event if $x\in A$ and $y\le x$ imply $y\in A$. Its complement is an increasing event. In bond percolation, with an open edge coded by $1$, closing more edges cannot destroy a decreasing event.

For example, the event that no open path connects two fixed vertices is decreasing; the event that such a path exists is increasing. Two decreasing events are positively associated under a product measure, since both of their complements are increasing events. An increasing event and a decreasing event instead satisfy the negative correlation of increasing and decreasing events. The order convention must be stated when the coordinates encode something other than open edges.