Perron-Wright asymptotics and critical component processes for regular inhomogeneous random graphs
Abstract
We extend Wright's enumeration of connected labelled graphs with fixed surplus to irreducible critical finite type kernels and regular compact type kernels. Let Ct(k,r) denote the total weight of connected labelled graphs with k vertices and surplus r. Let u>0 satisfy Ku=u and ∫ u dμ=1, and put α=∫ u2 dμ and β=∫ u3 dμ. We prove that the rooted-tree radius ρ is positive and real analytic near $1$, with ρ(1)=1 and logρ(t)=α2/2β(t−1)2+O(|t−1|3), and that, for fixed r∈ℤ≥0 and 0≤ L < ∞, Ct(k,r)=wrβ(r−1)/2e−kρ(t)−kkk+3r/2−2{1+Or,L(k−1/2)}, uniformly for |t−1|≤ Lk−1/2, where wr is the classical Wright constant. At tn=1+λn−1/3, these asymptotics yield local size and surplus point-process limits for deterministic finite type models and annealed compact type Bollobas-Janson-Riordan models satisfying projective quadratic regularity. After rescaling component sizes by β1/3n−2/3, the limit is the Aldous process at parameter αβ−2/3λ, and the corresponding type profiles concentrate at the Perron law p(dx)=u(x)μ(dx). For every regular countable critical kernel, we construct graph sequences with growing finite supports and the same local limits, and identify the additive shift of the critical window caused by spectral perturbations of order n−1/3.
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BibTeX
@misc{hu2026perronwright,
title = {{Perron-Wright asymptotics and critical component processes for regular inhomogeneous random graphs}},
author = {Wentao Hu and Wen Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23003},
eprint = {2610.23003},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23003},
}
RIS
TY - UNPB
TI - Perron-Wright asymptotics and critical component processes for regular inhomogeneous random graphs
AU - Wentao Hu
AU - Wen Sun
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23003
UR - https://arxiv.org/abs/2610.23003
AB - We extend Wright's enumeration of connected labelled graphs with fixed surplus to irreducible critical finite type kernels and regular compact type kernels. Let C_t(k,r) denote the total weight of connected labelled graphs with k vertices and surplus r. Let u>0 satisfy Ku=u and ∫ u dμ=1, and put α=∫ u^2 dμ and β=∫ u^3 dμ. We prove that the rooted-tree radius ρ is positive and real analytic near $1$, with ρ(1)=1 and logρ(t)=α^2/2β(t−1)^2+O(|t−1|^3), and that, for fixed r∈ℤ_≥0 and 0≤ L < ∞, C_t(k,r)=w_rβ^(r−1)/2e^−kρ(t)^−kk^k+3r/2−2{1+O_r,L(k^−1/2)}, uniformly for |t−1|≤ Lk^−1/2, where w_r is the classical Wright constant. At t_n=1+λn^−1/3, these asymptotics yield local size and surplus point-process limits for deterministic finite type models and annealed compact type Bollobas-Janson-Riordan models satisfying projective quadratic regularity. After rescaling component sizes by β^1/3n^−2/3, the limit is the Aldous process at parameter αβ^−2/3λ, and the corresponding type profiles concentrate at the Perron law p(dx)=u(x)μ(dx). For every regular countable critical kernel, we construct graph sequences with growing finite supports and the same local limits, and identify the additive shift of the critical window caused by spectral perturbations of order n^−1/3.
ER -