Lipschitz correspondence between metric measure spaces and random distance matrices

TL;DR

This paper establishes a bi-Lipschitz correspondence between metric measure spaces and distributions of finite random distance matrices, providing an effective version of Vershik's result that these spaces are determined by infinite random matrices.

Contribution

It introduces an asymptotically bi-Lipschitz relation linking metric measure spaces to finite random distance matrices, extending Vershik's theoretical framework.

Findings

Asymptotic bi-Lipschitz relation between spaces and matrices

Effective version of Vershik's determination result

Connection between metric measure spaces and random matrices

Abstract

Given a metric space with a Borel probability measure, for each integer $N$ we obtain a probability distribution on $N\times N$ distance matrices by considering the distances between pairs of points in a sample consisting of $N$ points chosen indepenedently from the metric space with respect to the given measure. We show that this gives an asymptotically bi-Lipschitz relation between metric measure spaces and the corresponding distance matrices. This is an effective version of a result of Vershik that metric measure spaces are determined by associated distributions on infinite random matrices.