Breaking of ensemble equivalence for perturbed Erd\H{o}s-R\'enyi random graphs

In [18] we analysed a simple undirected random graph subject to constraints on the total number of edges and the total number of triangles. We considered the dense regime in which the number of edges per vertex is proportional to the number of vertices. We showed that, as soon as the constraints are \emph{frustrated}, i.e., do not lie on the Erd\H{o}s-R\'enyi line, there is breaking of ensemble equivalence, in the sense that the specific relative entropy per edge of the \emph{microcanonical ensemble} with respect to the \emph{canonical ensemble} is strictly positive in the limit as the number of vertices tends to infinity. In the present paper we analyse what happens near the Erd\H{o}s-R\'enyi line. It turns out that the way in which the specific relative entropy tends to zero depends on whether the total number of triangles is slightly larger or slightly smaller than typical. We investigate what the constrained random graph looks like asymptotically in the microcanonical ensemble.