Scaling limit of dynamical percolation on critical Erdös-Rényi random graphs

The background subsection in the introduction was significantly expanded.The organization has changed, the main results are highlighted in section 3, some technical results are postponed in section 4 in order to have the main results stated earlier and other technical results are postponed to an appendix. Furthermore, a section dedicated to the perspectives has been added.Some important mistakes have been corrected:- the inappropriate use of compact convergence for stochastic processes, which causes measurability issues. Now, the usual Skorokhod topology is used (cf. appendix A).- the fact that even if a m.s-m.s is a collection of real graphs in $\mathcal{S}$, it is not necessarily the case that after a positive time, its coalescent remains a graph (for instance, one component might have infinite surplus), whence the introduction of the class $\mathcal{S}^{length}$ in section 2.6. There was thus a mistake in Lemma 4.10 (iii) (old numbering), corrected now in Lemma 5.9 (new numbering). See also the remarks after Theorem 3.3.- there was an error, now fixed, concerning the control of supdiam in section 5.4.- Lemma 2.5 (old numbering) as stated was false, a corrected version replaces it.Consider a critical Erdös-Rényi random graph: $n$ is the number of vertices, each one of the $\binom{n}{2}$ possible edges is kept in the graph independently from the others with probability $n^{−1}$ + $\lambda n^{−4/3}$ , $\lambda$ being a fixed real number. When $n$ goes to infinity, Addario-Berry, Broutin and Goldschmidt have shown that the collection of connected components, viewed as suitably normalized measured compact metric spaces, converges in distribution to a continuous limit $G_\lambda$ made of random real graphs. In this paper, we consider notably the dynamical percolation on critical Erdös-Rényi random graphs. To each pair of vertices is attached a Poisson process of intensity $n^{−1/3}$, and every time it rings, one resamples the corresponding edge. Under this process, the collection of connected components undergoes coalescence and fragmentation. We prove that this process converges in distribution, as $n$ goes to infinity, towards a fragmentation-coalescence process on the continuous limit $G_\lambda$. We also prove convergence of discrete coalescence and fragmentation processes and provide Feller-type properties associated to fragmentation and coalescence