  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 renormalisation. 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.
