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Strict Inequality for the Chemical Distance Exponent in Two‐Dimensional Critical Percolation
Author(s) -
Damron Michael,
Hanson Jack,
Sosoe Philippe
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21945
Subject(s) - mathematics , exponent , bounded function , upper and lower bounds , inverse , critical exponent , percolation critical exponents , percolation threshold , percolation (cognitive psychology) , combinatorics , triangle inequality , discrete mathematics , mathematical analysis , scaling , geometry , physics , quantum mechanics , philosophy , linguistics , neuroscience , biology , electrical resistivity and conductivity
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 ℤ 2 is bounded by Cn 2 − δ π 3 ( n ) for some δ > 0 , where π 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. © 2020 Wiley Periodicals LLC