We study cluster sizes in supercritical d-dimensional inhomogeneous percolation models with long-range edges —such as long-range percolation— and/or heavy-tailed degree distributions —such as geometric inhomogeneous random graphs and the age-dependent random connection model. Our focus is on large deviations of the size of the largest cluster in the graph restricted to a finite box as its volume tends to infinity. Compared to nearest neighbor Bernoulli bond percolation on <InlineEquation ID="IEq1"> <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>, we show that long edges can increase the exponent of the polynomial speed of the lower tail from <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(d-1)/d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>d</mi> </mrow> </math></EquationSource> </InlineEquation> to any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\zeta _\star \in \big ((d-1)/d,1\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mo>⋆</mo> </msub> <mo>∈</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>d</mi> <mo>,</mo> <mn>1</mn> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We prove that this exponent <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\zeta _\star $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ζ</mi> <mo>⋆</mo> </msub> </math></EquationSource> </InlineEquation> also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {C}(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. For the upper tail of large deviations, we prove that its speed is logarithmic for models with power-law degree distributions. We express the rate function via the generating function of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$|\mathcal {C}(0)|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.

Large deviations of the giant in supercritical kernel-based spatial random graphs

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2025年11月14日

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Joost Jorritsma,
Júlia Komjáthy,
Dieter Mitsche

摘要

We study cluster sizes in supercritical d-dimensional inhomogeneous percolation models with long-range edges —such as long-range percolation— and/or heavy-tailed degree distributions —such as geometric inhomogeneous random graphs and the age-dependent random connection model. Our focus is on large deviations of the size of the largest cluster in the graph restricted to a finite box as its volume tends to infinity. Compared to nearest neighbor Bernoulli bond percolation on $\mathbb {Z}^d$ $Z^{d}$ , we show that long edges can increase the exponent of the polynomial speed of the lower tail from $(d-1)/d$ $(d - 1) / d$ to any $\zeta _\star \in \big ((d-1)/d,1\big )$ $ζ_{⋆} \in ((d - 1) / d, 1)$ . We prove that this exponent $\zeta _\star $ $ζ_{⋆}$ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin $\mathcal {C}(0)$ $C (0)$ . For the upper tail of large deviations, we prove that its speed is logarithmic for models with power-law degree distributions. We express the rate function via the generating function of $|\mathcal {C}(0)|$ $| C (0) |$ . The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.