The lowest crossing in 2D critical percolation

We study the following problem for critical site percolation on the triangular lattice. Let $A$ and $B$ be sites on a horizontal line $e$ separated by distance $n$. Consider, in the half-plane above $e$, the lowest occupied crossing $R_n$ from the half-line left of $A$ to the half-line right of $B$. We show that the probability that $R_n$ has a site at distance smaller than $m$ from $AB$ is of order $(log (n/m))^{-1}$, uniformly in $1 leq m leq n/2$. Much of our analysis can be carried out for other two-dimensional lattices as well.