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

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 ${\mathbb{Z}}^d$, we show that long edges can increase the exponent of the polynomial speed of the lower tail from $(d-1)\mathrm{/}d$ to any ${\zeta }_{\star }\in ((d-1)\mathrm{/}d,1)$. We prove that this exponent ${\zeta }_{\star }$ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin $\mathcal{C}(0)$. 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 $\mathrm{\mid }\mathcal{C}(0)\mathrm{\mid }$. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.