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We provide the first nontrivial upper bound for the chemical distanceexponent in two-dimensional critical percolation. Specifically, we prove thatthe expected length of the shortest horizontal crossing path of a box of sidelength $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by$Cn^{2-\delta}\pi_3(n)$, for some $\delta>0$, where $\pi_3(n)$ is the"three-arm probability to distance $n$." This implies that the ratio of thislength to the length of the lowest crossing is bounded by an inverse power of$n$ with high probability. In the case of site percolation on the triangularlattice, we obtain a strict upper bound for the exponent of $4/3$.  The proof builds on the strategy developed in our previous paper, but with anew iterative scheme, and a new large deviation inequality for events in annuliconditional on arm events, which may be of independent interest.

Strict inequality for the chemical distance exponent in two-dimensional critical percolation