Abstract
We give a self-contained and detailed presentation of Kesten's results that allow to relate critical and near-critical percolation on the triangular lattice. They constitute an important step in the derivation of the exponents describing the near-critical behavior of this model. For future use and reference, we also show how these results can be obtained in more general situations, and we state some new consequences.
Highlights
Since 2000, substantial progress has been made on the mathematical understanding of percolation on the triangular lattice
Its mean density can be measured via the probability θ(p) that a given site belongs to this infinite black component
We would like to mention that these estimates for critical and near-critical percolation remain valid on other lattices too, like the square lattice – at least for the color sequences that we have used in the proofs, no analog of the color exchange trick being available
Summary
Since 2000, substantial progress has been made on the mathematical understanding of percolation on the triangular lattice. Reading Kesten’s paper in order to extract the statement that is needed to derive this result can turn out to be not so easy for a non-specialist, and the first goal of the present paper is to give a complete self-contained proof of Kesten’s results that are used to describe near-critical percolation. Other new statements in the present paper concern arms “with defects” or the fact that the finite-size scaling characteristic length Lǫ(p) remains of the same order of magnitude when ǫ varies in (0, 1/2) (Corollary 35) – and for ǫ small enough. This last fact is used in [39] to study the “off-critical” regime for percolation
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