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In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices n. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as n grows large, where two ensembles are said to be equivalent in the dense regime if this relative entropy divided by n 2 tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are frustrated. Examples are provided for three different choices of constraints.

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This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.

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