  Random graph models with limited choice have been studied extensively with
the goal of understanding the mechanism of the emergence of the giant
component. One of the standard models are the Achlioptas random graph processes
on a fixed set of $n$ vertices. Here at each step, one chooses two edges
uniformly at random and then decides which one to add to the existing
configuration according to some criterion. An important class of such rules are
the bounded-size rules where for a fixed $K\geq 1$, all components of size
greater than $K$ are treated equally. While a great deal of work has gone into
analyzing the subcritical and supercritical regimes, the nature of the critical
scaling window, the size and complexity (deviation from trees) of the
components in the critical regime and nature of the merging dynamics has not
been well understood. In this work we study such questions for general
bounded-size rules. Our first main contribution is the construction of an
extension of Aldous's standard multiplicative coalescent process which
describes the asymptotic evolution of the vector of sizes and surplus of all
components. We show that this process, referred to as the standard augmented
multiplicative coalescent (AMC) is `nearly' Feller with a suitable topology on
the state space. Our second main result proves the convergence of suitably
scaled component size and surplus vector, for any bounded-size rule, to the
standard AMC. The key ingredients here are a precise analysis of the asymptotic
behavior of various susceptibility functions near criticality and certain
bounds from [8], on the size of the largest component in the barely subcritical
regime.
