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.

中文翻译：

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声子，排列和Thoma单形：三个模高斯模空间

在本文中，我们展示了如何使用mod-Gaussian收敛框架来研究某些随机图模型，随机排列和随机整数分区的波动。我们证明，在这三个框架中，在适当的重归一化下，通用随机模型的通用同类可观测值是mod-Gaussian。这意味着中心极限定理具有正态性的扩展区域，中度偏差原理，收敛速度的估计，局部极限定理和浓度不等式。这些模型的可观测性的普遍渐近行为引起了模高斯模空间的概念。