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 Zᵈ, we show that long edges can increase the exponent of the polynomial speed of the lower tail from (d-1) /d to any _ ( (d-1) /d, 1). We prove that this exponent _ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin 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 |C (0) |. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.