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Strict inequalities for arm exponents in planar percolation
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DOI：10.1007/s00440-025-01448-8.png)
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Abstract
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We discuss a general method to obtain quantitative improvements of correlation inequalities and apply it to arm estimates for Bernoulli bond percolation on Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}<^>2$$\end{document}. 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.
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