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Strict inequality for the chemical distance exponent in two-dimensional critical percolation
- Publication Year :
- 2017
-
Abstract
- 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.<br />Comment: 45 pages, 7 figures
- Subjects :
- Mathematics - Probability
60K35, 82B43
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1708.03643
- Document Type :
- Working Paper