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 foun...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
21.01.2020
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| Subjects | |
| Online Access | Get full text |
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| Summary: | 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 $\gamma$
from the origin to distance $n$ that consists of "three-arm" points, and whose
volume can thus be estimated by $O(n^2\pi_3(n))$. Here, $\pi_3(n)$ 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 $\{e\in \gamma\}$,
to obtain a bound of the form $O(n^{2-\delta}\pi_3(n))$ for some $\delta>0$. |
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| DOI: | 10.48550/arxiv.2001.07872 |