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In this paper, we show how to use the framework of mod‐Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod‐Gaussian under an appropriate renormalization. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod‐Gaussian moduli space.

Graphons, permutons and the Thoma simplex: three mod‐Gaussian moduli spaces