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

Abstract 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 $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , we show that long edges can increase the exponent of the polynomial speed of the lower tail from $$(d-1)/d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:math> to any $$\zeta _\star \in \big ((d-1)/d,1\big )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We prove that this exponent $$\zeta _\star $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mo>⋆</mml:mo> </mml:msub> </mml:math> also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin $$\mathcal {C}(0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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 $$|\mathcal {C}(0)|$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> . The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear .