Abstract
AbstractWe 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 ℤ2 is bounded by Cn2 − δπ3(n) for some δ > 0, where π3(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. © 2020 Wiley Periodicals LLC
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
