The mutual affinity of random measures

TL;DR

This paper studies the spectral properties of Gram matrices formed from random probability measures on finite spaces, showing convergence of eigenvalue distributions as the system size grows.

Contribution

It introduces the concept of mutual affinity via Gram matrix spectra and proves convergence of eigenvalue distributions for large random measures.

Findings

Eigenvalue distribution converges to a fixed limit for large systems.

The mutual affinity relates to the spectral properties of Gram matrices.

The results apply as both the number of measures and event space size grow.

Abstract

We consider a set of probability measures on a finite event space $\Omega$. The mutual affinity is introduced in terms of the spectrum of the associated Gram matrix. We show that, for randomly chosen measures, the empirical eigenvalue distribution of the Gram matrix converges to a fixed distribution in the limit where the number of measures, together with the cardinality of $\Omega$, goes to infinity.