Year: 2025
Type: article
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 $$\mathbb {Z}^d$$ 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 $$\zeta _\star \in \big ((d-1)/d,1\big )$$ ζ ⋆ ∈ ( ( d - 1 ) / d , 1 ) . We prove that this exponent $$\zeta _\star $$ ζ ⋆ also governs the size of the second-largest cluster, and the distribution of the size of the cluster containing the origin $$\mathcal {C}(0)$$ 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 $$|\mathcal {C}(0)|$$ | C ( 0 ) | . The upper tail in degree-homogeneous models decays much faster: the speed in long-range percolation is linear .
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4416222091