Near critical asymptotics in the frozen Erdős-Rényi

We consider a modification of the standard Erds-Rnyi random graph, where the growth of components containing a unique cycle is slowed down and the apparition of multicyclic components is forbidden.It was proven by Contat and Curien (2023) that the sizes of the connected components of this graph undergo a similar phase transition to that of the classical model.In particular, in the critical window the scaling limit of the sizes of the components is a "frozen" version of Aldous' multiplicative coalescent (Aldous, 1997).Consequently the total number of vertices belonging to a cyclic component also converges after suitable renormalization.The aim of this article is to study the process arising as this scaling limit.More precisely, we describe its long time asymptotics in the critical window.