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One-arm exponents of high-dimensional percolation revisited

Diederik van Engelenburg, Christophe Garban, Romain Panis, Franco Severo

TL;DR

The paper proves, for $d>6$ and large spread $L$, that the one-arm probability at criticality decays as $\mathbb{P}_{p_c}[0\leftrightarrow\Lambda_n^c] \asymp n^{-2}$ in full-space and $\mathbb{P}_{p_c}[0\leftrightarrow_{\mathbb{H}}\Lambda_n^c] \asymp n^{-3}$ in the half-space for spread-out percolation. It achieves this with a streamlined, lace-expansion-free approach that hinges on the entropic bound of Dewan and Muirhead, together with sharp near-critical inputs for the susceptibility and a novel correlation-length notion $L(p)$ (and $\tilde L(p)$) from Dumínil-Copin and Panis. The method yields a robust, self-contained proof of the Kozma–Nachmias result in full-space and the Chatterjee–Hanson half-space exponent, and it also underpins a complementary half-space two-point function analysis, all while aligning with related high-dimensional Ising model techniques. This provides a practical, adaptable framework for mean-field-type exponent calculations in high-dimensional percolation and related models, without recourse to lace expansions.

Abstract

We consider sufficiently spread-out Bernoulli percolation in dimensions ${d>6}$. We present a short and simple proof of the up-to-constants estimate for the one-arm probability in both the full-space and half-space settings. These results were previously established by Kozma and Nachmias and by Chatterjee and Hanson, respectively. Our proof improves upon the entropic technique introduced by Dewan and Muirhead, relying on a sharp estimate on a suitably chosen correlation length recently obtained by Duminil-Copin and Panis. This approach is inspired by our companion work, where we compute the one-arm exponent for several percolation models related to the high-dimensional Ising model.

One-arm exponents of high-dimensional percolation revisited

TL;DR

The paper proves, for and large spread , that the one-arm probability at criticality decays as in full-space and in the half-space for spread-out percolation. It achieves this with a streamlined, lace-expansion-free approach that hinges on the entropic bound of Dewan and Muirhead, together with sharp near-critical inputs for the susceptibility and a novel correlation-length notion (and ) from Dumínil-Copin and Panis. The method yields a robust, self-contained proof of the Kozma–Nachmias result in full-space and the Chatterjee–Hanson half-space exponent, and it also underpins a complementary half-space two-point function analysis, all while aligning with related high-dimensional Ising model techniques. This provides a practical, adaptable framework for mean-field-type exponent calculations in high-dimensional percolation and related models, without recourse to lace expansions.

Abstract

We consider sufficiently spread-out Bernoulli percolation in dimensions . We present a short and simple proof of the up-to-constants estimate for the one-arm probability in both the full-space and half-space settings. These results were previously established by Kozma and Nachmias and by Chatterjee and Hanson, respectively. Our proof improves upon the entropic technique introduced by Dewan and Muirhead, relying on a sharp estimate on a suitably chosen correlation length recently obtained by Duminil-Copin and Panis. This approach is inspired by our companion work, where we compute the one-arm exponent for several percolation models related to the high-dimensional Ising model.
Paper Structure (14 sections, 11 theorems, 88 equations, 1 figure)

This paper contains 14 sections, 11 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and statement of the results
  3. Notations.
  4. Strategy of the proof
  5. Acknowledgements.
  6. Preliminaries
  7. Bounds on $\varphi_p$
  8. The sharp length
  9. Bounds on the half-space susceptibility
  10. One-arm probability in the full-space
  11. One-arm probability in the half-space
  12. Proof of the lower bound
  13. Proof of the upper bound
  14. Percolation density, volume and magnetisation

Key Result

Theorem 1.1

Let $d>6$. There exists $L_0\geq 1$ such that the following holds. For every $L\geq L_0$, there exist $c,C>0$ such that, for every $n\geq 1$,

Figures (1)

  • Figure 1: An illustration of the decomposition used in \ref{['eq:final step up hs']}

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.3: DewanMuirhead
  • proof
  • Theorem 1.4: DumPan24Perco
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 16 more