Fluctuations of the Gromov-Prohorov sample model

TL;DR

This paper investigates the fluctuation behavior of polynomial observables in random metric measure spaces, revealing normal fluctuations for generic spaces and non-Gaussian limits for compact homogeneous spaces, with implications for space classification.

Contribution

It characterizes the fluctuation patterns of observables in random metric measure spaces, distinguishing between generic and compact homogeneous spaces through asymptotic distribution analysis.

Findings

Generic spaces exhibit asymptotically normal fluctuations.

Compact homogeneous spaces have smaller, non-Gaussian fluctuations.

Fluctuation size characterizes the space's homogeneity.

Abstract

In this paper, we study the fluctuations of observables of metric measure spaces which are random discrete approximations $X_n$ of a fixed arbitrary (complete, separable) metric measure space $X=(\mathcal{X},d,\mu)$. These observables $\Phi(X_n)$ 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 $\Phi(X_n)$ 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 F\'eray-M\'eliot-Nikeghbali.