A conditional compound Poisson process approach to the sparse Erd\H{o}s-R\'enyi random graphs: moderate deviations

TL;DR

This paper introduces a novel compound Poisson process model for analyzing the sizes of connected components in sparse Erdős-Rényi graphs, linking phase transitions to condensation phenomena and deriving moderate deviation principles.

Contribution

It presents a new representation of component sizes using a conditioned compound Poisson process, connecting phase transitions with condensation in zero-range models.

Findings

Derived moderate deviation principles for component sizes and counts

Established connections between phase transition and condensation phenomena

Discussed large deviation results for the model

Abstract

We construct a compound Poisson process conditioned on its random summation that represents the sizes of the connected components in the sparse Erd\H{o}s-R\'enyi random graph $G(n,c/n)$. This new representation depicts a connection between the phase transition in the sparse random graph and the condensation transition in the zero-range model. Under this framework, we can derive moderate deviation principles for the maximun component, total number of connected components and empirical measure of the sizes in the non-critical regimes. Large deviation results are discussed.