Near-critical percolation in two dimensions

TL;DR

This paper provides a detailed presentation of Kesten's results linking critical and near-critical percolation on the triangular lattice, advancing understanding of near-critical behavior and extending applicability to more general cases.

Contribution

It offers a comprehensive exposition of Kesten's results and demonstrates their broader relevance and new implications in percolation theory.

Findings

Relates critical and near-critical percolation on the triangular lattice

Derives exponents for near-critical behavior

Extends results to more general situations

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.