We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint distribution of the number of connected components, of the sizes of the giant components and of the numbers of the excess edges of the giant components. For the supercritical case, we obtain the asymptotics of normal deviations and the logarithmic asymptotics of large and moderate deviations of the joint distribution of the number of components, of the size of the largest component and of the number of the excess edges of the largest component. For the critical case, we obtain the logarithmic asymptotics of moderate deviations of the joint distribution of the sizes of connected components and of the numbers of the excess edges. Some related asymptotics are also established. The proofs of the large and moderate deviation asymptotics employ methods of idempotent probability theory. As a byproduct of the results, we provide some additional insight into the nature of phase transitions in sparse random graphs.
"Stochastic processes in random graphs." Ann. Probab. 33 (1) 337 - 412, January 2005. https://doi.org/10.1214/009117904000000784