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

TL;DR

This paper investigates the large deviation behavior of the largest cluster in supercritical inhomogeneous percolation models with long-range edges or heavy-tailed degree distributions, revealing how these features influence deviation probabilities.

Contribution

It provides new insights into how long edges and heavy tails affect the large deviation exponents of cluster sizes in supercritical spatial random graphs.

Findings

Long edges increase the lower tail exponent from (d-1)/d to any value in ((d-1)/d,1).

The same exponent governs the size of the second-largest cluster and the cluster containing the origin.

Upper tail deviations decay faster in degree-homogeneous models, with a linear speed in long-range percolation.

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 $\mathbb{Z}^d$, we show that long edges can increase the exponent of the polynomial speed of the lower tail from $(d-1)/d$ to any $\zeta_\star\in\big((d-1)/d,1\big)$. 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 $|\mathcal{C}(0)|$. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.