Perron--Wright asymptotics and critical component processes for inhomogeneous random graphs

Abstract

We extend Wright's enumeration of connected labelled graphs with fixed surplus to irreducible critical finite type kernels and continuous compact type kernels satisfying a positivity condition. Let ${\mathsf{C}}_t(k,r)$ denote the total weight of connected labelled graphs with $k$ vertices and surplus $r$. Let $u>0$ satisfy $Ku=u$ and $\int ud\mu =1$, and put $\alpha =\int u^{2}d\mu$ and $\beta =\int u^{3}d\mu$. We prove that the rooted-tree radius $\rho$ is positive and real analytic near $1$, with $\rho (1)=1$ and $\log \rho (t)=\frac{{\alpha }^2}{2\beta }(t-1{)}^2+O(|t-1{|}^3)$, and that, for fixed $r\in {\mathbb{Z}}_{\ge 0}$ and $0\le L<\infty$, ${\mathsf{C}}_t(k,r)=w_r{\beta }^{(r-1)/2}{e}^{-k}\rho (t{)}^{-k}{k}^{k+3r/2-2}\{1+O_{r,L}(k^{-1/2})\}$, uniformly for $|t-1|\le L{k}^{-1/2}$, where $w_r$ is the classical Wright constant. At criticality, the leading constant depends on the kernel only through $\beta$, under the stated normalization of $u$. At $t_n=1+\lambda n^{-1/3}$, exact component identities combine with these asymptotics to give fixed size factorial densities and local size and surplus point-process limits for deterministic finite type models and annealed compact type Bollobás-Janson-Riordan models. With sizes rescaled by ${\beta }^{1/3}{n}^{-2/3}$, the limit is the Aldous process at parameter $\alpha {\beta }^{-2/3}\lambda$, and the corresponding type profiles concentrate at the Perron law $p(dx)=u(x)\mu (dx)$. For every bounded uniformly positive 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 $n^{-1/3}$.

arXiv comment58 pages, 3 figures