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

Abstract This article proposes a new way of deriving mean-field exponents for sufficiently spread-out Bernoulli percolation in dimensions $$d&gt;6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> . 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 $$d&gt;4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> [Duminil-Copin and Panis, arXiv:2410.03649 , (2024)].