We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ 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 this length 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 triangular lattice, 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 a new iterative scheme, and a new large deviation inequality for events in annuli conditional on arm events, which may be of independent interest.

中文翻译：

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二维临界渗流中化学距离指数的严格不等式

我们为二维临界渗透中的化学距离指数提供了第一个非平凡的上限。具体来说，我们证明了在 $\mathbb{Z}^2$ 上的临界渗流中边长 $n$ 的盒子的最短水平交叉路径的期望长度以 $Cn^{2-\delta}\pi_3 为界(n)$，对于某些 $\delta>0$，其中 $\pi_3(n)$ 是“距离 $n$ 的三臂概率”。这意味着该长度与最低交叉点的长度的比率以高概率受到 $n$ 的逆幂的限制。在三角形格子上的站点渗透的情况下，我们获得了 $4/3$ 指数的严格上限。证明建立在我们之前论文中开发的策略之上，但使用了新的迭代方案，以及以 arm 事件为条件的环中事件的新大偏差不等式，