Near critical scaling relations for planar Bernoulli percolation without differential inequalities

TL;DR

This paper offers a new, simpler proof of a key scaling relation in Bernoulli percolation on the square lattice, avoiding differential inequalities and Russo's formula, with potential applicability to other models.

Contribution

It introduces a novel approach that relates crossing probability differences at various scales, providing a more robust proof of the near-critical scaling relation.

Findings

New proof of the scaling relation $eta=\xi_1 u$

Method avoids Russo's formula and differential inequalities

Potential to extend approach to other scaling relations

Abstract

We provide a new proof of the near-critical scaling relation $\beta=\xi_1\nu$ for Bernoulli percolation on the square lattice already proved by Kesten in 1987. We rely on a novel approach that does not invoke Russo's formula, but rather relates differences in crossing probabilities at different scales. The argument is shorter and more robust than previous ones and is more likely to be adapted to other models. The same approach may be used to prove the other scaling relations appearing in Kesten's work.