# A large-deviations principle for all the cluster sizes of a sparse Erd\H{o}s-R\'enyi graph

## TL;DR

This paper establishes a large-deviations principle for the distribution of all component sizes in a sparse Erdős-Rényi graph, capturing phase transition phenomena and linking to coagulation models.

## Contribution

It provides an explicit rate function describing microscopic, mesoscopic, and macroscopic component sizes, including the phase transition at t=1.

## Key findings

- Explicit large-deviations rate function for component sizes.
- Captures phase transition at t=1.
- Links to coagulation models and gelation phenomena.

## Abstract

Let $\mathcal{G}(N,\frac 1Nt_N)$ be the Erd\H{o}s-R\'enyi graph with connection probability $\frac 1Nt_N\sim t/N$ as $N\to\infty$ for a fixed $t\in(0,\infty)$. We derive a large-deviations principle for the empirical measure of the sizes of all the connected components of $\mathcal{G}(N,\frac 1Nt_N)$, registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order $N$), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic and macroscopic components and the fraction of vertices in components of mesoscopic sizes. Moreover, it clearly captures the well known phase transition at $t=1$ as part of a comprehensive picture. The proofs rely on elementary combinatorics and on known estimates and asymptotics for the probability that subgraphs are connected. We also draw conclusions for the strongly related model of the multiplicative coalescent, the Marcus--Lushnikov coagulation model with monodisperse initial condition, and its gelation phase transition.