Strict inequalities for arm exponents in planar percolation

TL;DR

This paper introduces a new method to improve correlation inequalities in planar percolation, demonstrating that arm exponents are strictly larger than previously known bounds, with implications for dynamical percolation.

Contribution

It provides the first proof that the two-arm exponent exceeds twice the one-arm exponent, and shows monochromatic arm exponents are strictly larger than polychromatic ones, refining existing inequalities.

Findings

Two-arm exponent is strictly larger than twice the one-arm exponent.

Monochromatic arm exponents are strictly larger than polychromatic ones.

Quantitative improvements on Harris-FKG and Reimer's inequalities.

Abstract

We discuss a general method to prove quantitative improvements on correlation inequalities and apply it to arm estimates for Bernoulli bond percolation on the square lattice. Our first result is that the two-arm exponent is strictly larger than twice the one-arm exponent and can be seen as a quantitative improvement on the Harris-FKG inequality. This answers a question of Garban and Steif, which was motivated by the study of exceptional times in dynamical percolation. Our second result is that the monochromatic arm exponents are strictly larger than their polychromatic versions, and can be seen as a quantitative improvement on Reimer's main lemma. This second result is not new and was already proved by Beffara and Nolin using a different argument.