Law of addition in random matrix theory

TL;DR

This paper develops a diagrammatic method to analyze the spectral density of sums of deterministic and random matrices, generalizing previous results and applicable to various random matrix problems.

Contribution

It introduces the concept of 'gluon connectedness' and extends the calculation of energy level densities to a broader class of distributions.

Findings

Derived a generalized formula for energy level density

Introduced the concept of 'gluon connectedness' in random matrix addition

Applicable to a wide range of probability distributions

Abstract

We discuss the problem of adding random matrices, which enable us to study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of ``gluon connectedness," we calculate the density of energy levels for a wide class of probability distributions governing the random term, thus generalizing a result obtained recently by Br\'ezin, Hikami, and Zee. The method used here may be applied to a broad class of problems involving random matrices.