Strict inequalities for arm exponents in planar percolation

Abstract We discuss a general method to obtain quantitative improvements of correlation inequalities and apply it to arm estimates for Bernoulli bond percolation on $${\mathbb {Z}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . 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 of the Harris-FKG inequality. This answers a question of Garban and Steif [10, Open Problem 13.6], which was motivated by the study of exceptional times in dynamical percolation [24, section 9]. Our second result is that the monochromatic arm exponents are strictly larger than their polychromatic versions, and can be seen as a quantitative improvement of Reimer’s main lemma [1, Lemma 4.1]. This second result is not new; it was already proved by Beffara and Nolin [3] using a different argument.