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

TL;DR

This paper introduces a unified framework using mod-Gaussian convergence to analyze fluctuations in random graphs, permutations, and partitions, revealing universal asymptotic behaviors and establishing a new mod-Gaussian moduli space.

Contribution

It develops the concept of mod-Gaussian moduli space and applies it to three different probabilistic models, providing comprehensive fluctuation results and limit theorems.

Findings

Establishes mod-Gaussian behavior for generic observables

Derives central limit theorems with extended normality zones

Provides concentration inequalities and deviation principles

Abstract

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.