Large deviations of the giant in supercritical kernel-based spatial random graphs
arxiv(2024)
摘要
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 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 ζ_⋆∈((d-1)/d,1). We prove that this exponent
ζ_⋆ 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.
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