  We consider a variant of the classical Erd\H{o}s-R\'enyi random graph, where
components with surplus are slowed down to prevent the apparition of complex
components. The sizes of the components of this process undergo a similar phase
transition to that of the classical model, and in the critical window the
scaling limit of the sizes of the components is a "frozen" version of Aldous'
multiplicative coalescent [2]. The aim of this article is to describe the long
time asymptotics in the critical window for the total number of vertices which
belong to a component with surplus.
