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

Journal article

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$$\mathbb {Z}^d$$, 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 ζ⋆∈((d-1)/d,1)$$\zeta _\star \in \big ((d-1)/d,1\big )$$. We prove that this exponent ζ⋆$$\zeta _\star $$ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin C(0)$$\mathcal {C}(0)$$. 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 |C(0)|$$|\mathcal {C}(0)|$$. The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear.

Authors: Jorritsma, J; Komjáthy, J; Mitsche, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Probability Theory and Related Fields
• Pages: 1-100
• Publication date: 2025-11-14
• DOI: 10.1007/s00440-025-01417-1
• EISSN: 1432-2064
• ISSN: 0178-8051
• Language: English