# The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory

arXiv: https://arxiv.org/abs/2012.00663 · HTML: https://mlsanity.com/paper/2012.00663/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2020.

## Built on by (most cited first, 1 of 1)

- [Bias-variance decomposition of overparameterized regression with random linear features](https://mlsanity.com/paper/2203.05443/) (2022)

## Abstract (arXiv)

We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko-Pastur Law for Wishart matrices, the spectral density for a product Wishart matrix composed of two square matrices, and the Circle and elliptic laws for real random matrices.