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Random graph models with limited choice have been studied extensively withthe goal of understanding the mechanism of the emergence of the giantcomponent. One of the standard models are the Achlioptas random graph processeson a fixed set of $n$ vertices. Here at each step, one chooses two edgesuniformly at random and then decides which one to add to the existingconfiguration according to some criterion. An important class of such rules arethe bounded-size rules where for a fixed $K\geq 1$, all components of sizegreater than $K$ are treated equally. While a great deal of work has gone intoanalyzing the subcritical and supercritical regimes, the nature of the criticalscaling window, the size and complexity (deviation from trees) of thecomponents in the critical regime and nature of the merging dynamics has notbeen well understood. In this work we study such questions for generalbounded-size rules. Our first main contribution is the construction of anextension of Aldous's standard multiplicative coalescent process whichdescribes the asymptotic evolution of the vector of sizes and surplus of allcomponents. We show that this process, referred to as the standard augmentedmultiplicative coalescent (AMC) is `nearly' Feller with a suitable topology onthe state space. Our second main result proves the convergence of suitablyscaled component size and surplus vector, for any bounded-size rule, to thestandard AMC. The key ingredients here are a precise analysis of the asymptoticbehavior of various susceptibility functions near criticality and certainbounds from [8], on the size of the largest component in the barely subcriticalregime.

The augmented multiplicative coalescent and critical dynamic random graph models