Asymptotic Eigenvalue Distribution of Wishart Matrices whose Components are not Independently and Identically Distributed

TL;DR

This paper analyzes the asymptotic eigenvalue distribution of Wishart matrices with non-i.i.d. components using replica analysis and belief propagation, extending understanding beyond traditional i.i.d. assumptions.

Contribution

It introduces a novel approach combining replica analysis and belief propagation to determine eigenvalue distributions for non-i.i.d. Wishart matrices.

Findings

Supports results with Feynman diagram analysis

Provides an algorithm for eigenvalue distribution calculation

Validates methods through numerical experiments

Abstract

In the present work, eigenvalue distributions defined by a random rectangular matrix whose components are neither independently nor identically distributed are analyzed using replica analysis and belief propagation. In particular, we consider the case in which the components are independently but not identically distributed; for example, only the components in each row or in each column may be {identically distributed}. We also consider the more general case in which the components are correlated with one another. We use the replica approach while making only weak assumptions in order to determine the asymptotic eigenvalue distribution and to derive an algorithm for doing so, based on belief propagation. One of our findings supports the results obtained from Feynman diagrams. We present the results of several numerical experiments that validate our proposed methods.