1. Graphons, permutons and the Thoma simplex: three mod‐Gaussian moduli spaces
- Author
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Ashkan Nikeghbali, Valentin Féray, Pierre-Loïc Méliot, Universität Zürich [Zürich] = University of Zurich (UZH), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), University of Zurich, and Méliot, Pierre‐Loïc
- Subjects
Pure mathematics ,General Mathematics ,Gaussian ,340 Law ,610 Medicine & health ,0102 computer and information sciences ,01 natural sciences ,symbols.namesake ,510 Mathematics ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Limit (mathematics) ,0101 mathematics ,Concentration inequality ,ComputingMilieux_MISCELLANEOUS ,2600 General Mathematics ,Mathematics ,Central limit theorem ,Random graph ,Simplex ,Probability (math.PR) ,010102 general mathematics ,Observable ,10003 Department of Banking and Finance ,Moduli space ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,10123 Institute of Mathematics ,010201 computation theory & mathematics ,symbols ,Mathematics - Probability - 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., Comment: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures
- Published
- 2020