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<p>This article proposes a new way of deriving mean-field exponents for sufficiently spread-out Bernoulli percolation in dimensions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(d&gt;6\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>&gt;</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>. We obtain up-to-constant estimates for the full-space and half-space two-point functions in the critical and near-critical regimes. In a companion paper, we apply a similar analysis to the study of the weakly self-avoiding walk model in dimensions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(d&gt;4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>&gt;</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> [Duminil-Copin and Panis, <a href="http://arxiv.org/abs/2410.03649">arXiv:2410.03649</a>, (2024)].</p>

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An alternative approach for the mean-field behaviour of spread-out Bernoulli percolation in dimensions \(d>6\)

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  • 发表日期：

    2025年10月4日

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  • Hugo Duminil-Copin,
  • Romain Panis

摘要

This article proposes a new way of deriving mean-field exponents for sufficiently spread-out Bernoulli percolation in dimensions \(d>6\) d > 6 . We obtain up-to-constant estimates for the full-space and half-space two-point functions in the critical and near-critical regimes. In a companion paper, we apply a similar analysis to the study of the weakly self-avoiding walk model in dimensions \(d>4\) d > 4 [Duminil-Copin and Panis, arXiv:2410.03649, (2024)].