We investigate the bond percolation model on transient weighted graphs G induced by the excursion sets of the Gaussian free field on the corresponding metric graph. Under the sole assumption that its sign clusters do not percolate, we derive an extension of Lupu’s formula for the two-point function at criticality. We then focus on the low-dimensional case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0< \nu < \frac{\alpha }{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ν</mi> <mo><</mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> governs the polynomial volume growth of G and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation> the decay rate of the Green’s function on G. In particular, this includes the benchmark case <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${G}=\mathbb {Z}^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, for which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha =3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\nu = \alpha -2=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>=</mo> <mi>α</mi> <mo>-</mo> <mn>2</mn> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove under these assumptions that the critical one-arm probability decays with distance R like <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$R^{-\frac{\nu }{2}}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mrow> <mo>-</mo> <mfrac> <mi>ν</mi> <mn>2</mn> </mfrac> </mrow> </msup> </math></EquationSource> </InlineEquation>, up to multiplicative constants.

Critical one-arm probability for the metric Gaussian free field in low dimensions

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2025年7月15日

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Alexander Drewitz,
Alexis Prévost,
Pierre-François Rodriguez

摘要

We investigate the bond percolation model on transient weighted graphs G induced by the excursion sets of the Gaussian free field on the corresponding metric graph. Under the sole assumption that its sign clusters do not percolate, we derive an extension of Lupu’s formula for the two-point function at criticality. We then focus on the low-dimensional case $0< \nu < \frac{\alpha }{2}$ $0 < ν < \frac{α}{2}$ , where $\alpha $ $α$ governs the polynomial volume growth of G and $\nu $ $ν$ the decay rate of the Green’s function on G. In particular, this includes the benchmark case ${G}=\mathbb {Z}^3$ $G = Z^{3}$ , for which $\alpha =3$ $α = 3$ and $\nu = \alpha -2=1$ $ν = α - 2 = 1$ . We prove under these assumptions that the critical one-arm probability decays with distance R like $R^{-\frac{\nu }{2}}$ $R^{- \frac{ν}{2}}$ , up to multiplicative constants.