Abstract
In this paper, we study the fluctuations of observables of metric measure spaces which are random discrete approximations Xn of a fixed arbitrary (complete, separable) metric measure space X=(X,d,μ). These observables Φ(Xn) are polynomials in the sense of Greven–Pfaffelhuber–Winter, and we show that for a generic model space X, they yield asymptotically normal random variables. However, if X is a compact homogeneous space, then the fluctuations of the observables are much smaller, and after an adequate rescaling, they converge towards probability distributions which are not Gaussian. Conversely, we prove that if all the fluctuations of the observables Φ(Xn) are smaller than in the generic case, then the measure metric space X is compact homogeneous. The proofs of these results rely on the Gromov reconstruction principle, and on an adaptation of the method of cumulants and mod-Gaussian convergence developed by Féray–Méliot–Nikeghbali. As an application of our results, we construct a statistical test of the hypothesis of symmetry of a compact Riemannian manifold.
Highlights
Let X = (X, d, μ) be a metric space which we assume to be complete, separable and equipped with a probability measure μ over the Borel algebra of X ; and (Xn)n∈N be a sequence of independent random variables with the same law μ
We study here the approximation of X = (X, d, μ) by the random discrete metric space
The Gromov-weak topology is based on the idea that a sequence
Summary
In a second part, we study the case where the variance of Φ(Xn) is at most of order 1/n2 for any polynomial Φ We call this setting a globally singular point X of the Gromov–Prohorov sample model. We believed that the singular points of a mod-Gaussian moduli space still yielded observables which were asymptotically normal, albeit with a different rescaling This is what happens for singular graphons (Erdos–Rényi random graphs) and for singular models of random integer partitions (Plancherel and Schur–Weyl measures).
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