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An estimate for the radial chemical distance in $2d$ critical percolation clusters

We derive an estimate for the distance, measured in lattice spacings, inside two-dimensional critical percolation clusters from the origin to the boundary of the box of side length $2n$, conditioned on the existence of an open connection. The estimate we obtain is the radial analogue of the one found in the work of Damron, Hanson, and Sosoe. In the present case, however, there is no lowest crossing in the box to compare to, so we construct a path $γ$ from the origin to distance $n$ that consists of "three-arm" points, and whose volume can thus be estimated by $O(n^2π_3(n))$. Here, $π_3(n)$ is the "three-arm probability" that the origin is connected to distance $n$ by three arms, two open and one dual-closed. We then develop estimates for the existence of shortcuts around an edge $e$ in the box, conditional on $\{e\in γ\}$, to obtain a bound of the form $O(n^{2-δ}π_3(n))$ for some $δ>0$.