Universal Asymptotic Eigenvalue Distribution of Large $N$ Random Matrices --- A Direct Diagrammatic Proof to Marchenko-Pastur Law ---

TL;DR

This paper offers a new diagrammatic proof of the Marchenko-Pastur law in random matrix theory, demonstrating its universality and extending it to six types of restricted matrices, aiding physics research.

Contribution

It provides an alternative, diagrammatic proof of the Marchenko-Pastur law and generalizes it to additional classes of restricted random matrices.

Findings

Diagrammatic proof of Marchenko-Pastur law

Extension to six types of restricted matrices

Enhanced understanding for physics applications

Abstract

In random matrix theory, Marchenko-Pastur law states that random matrices with independent and identically distributed entries have a universal asymptotic eigenvalue distribution under large dimension limit, regardless of the choice of entry distribution. This law provides useful insight for physics research, because the large $N$ limit proved to be a very useful tool in various theoretical models. We present an alternative proof of Marchenko- Pastur law using Feynman diagrams, which is more familiar to the physics community. We also show that our direct diagrammatic approach can readily generalize to six types of restricted random matrices, which are not all covered by the original Marchenko-Pastur law.