A conditional compound Poisson process approach to the sparse Erdős-Rényi random graphs: moderate deviations

We construct a compound Poisson process conditioned on its random summation that represents the sizes of the connected components in the sparse Erdős-Rényi random graph $G(n,c\mathrm{/}n)$. 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.