On the block size spectrum of a class of exchangeable dynamic random graphs

In this work we introduce the dynamic $\Theta$-random graph and the associated $\Theta$-coalescent with momentum. Dynamic $\Theta$-random graphs are a subclass of exchangeable and consistent random graph processes, parametrised by a measure $\Theta$ on $[0,1]\times (0,1]$, inspired by the classic $\Lambda$-coalescent from mathematical population genetics. The $\Theta$-coalescent with momentum accounts for the small connected components of this graph; in contrast to the underlying random graph it is exchangeable but not consistent. Our main results specialise on the case where $\Theta$ is the product of a beta measure and a Dirac mass at $1$. We prove a dynamic law of large numbers for the block size spectrum, which tracks the numbers of blocks containing $1,...,d$ elements. On top of that, we provide a functional limit theorem for the fluctuations. The limit process satisfies a stochastic differential equation of Ornstein-Uhlenbeck type.