<p>We study a percolation model on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> called the random connection model. For <i>d</i> large, we use the lace expansion to prove that the critical two-point connection probability decays like <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$|x|^{-(d-2)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|x| \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d \ge 11$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation> and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> version of Hara’s induction argument.</p>

Decay of Connection Probability in High-Dimensional Continuum Percolation

期刊论文
发表日期：

2026年4月22日

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Matthew Dickson,
Yucheng Liu

摘要

We study a percolation model on $\mathbb {R}^d$ $R^{d}$ called the random connection model. For d large, we use the lace expansion to prove that the critical two-point connection probability decays like $|x|^{-(d-2)}$ ${| x |}^{- (d - 2)}$ as $|x| \rightarrow \infty $ $| x | \to \infty$ , with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on $\mathbb {Z}^d$ $Z^{d}$ in $d \ge 11$ $d \geq 11$ and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an $L^p$ $L^{p}$ version of Hara’s induction argument.