Strict inequality for the chemical distance exponent in twodimensional critical percolation
Abstract
We provide the first nontrivial upper bound for the chemical distance exponent in twodimensional 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 "threearm 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.03643
 arXiv:
 arXiv:1708.03643
 Bibcode:
 2017arXiv170803643D
 Keywords:

 Mathematics  Probability;
 60K35;
 82B43
 EPrint:
 45 pages, 7 figures