# Applications of Realizations (aka Linearizations) to Free Probability

arXiv: https://arxiv.org/abs/1511.05330 · HTML: https://mlsanity.com/paper/1511.05330/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Nov 2015.

## Built on by (most cited first, 8 of 8)

- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Denoising Score Matching with Random Features: Insights on Diffusion Models from Precise Learning Curves](https://mlsanity.com/paper/2502.00336/) (2025)
- [Model, sample, and epoch-wise descents: exact solution of gradient flow in the random feature model](https://mlsanity.com/paper/2110.11805/) (2021)
- [Demystifying Disagreement-on-the-Line in High Dimensions](https://mlsanity.com/paper/2301.13371/) (2023)
- [Gradient flow in the gaussian covariate model: exact solution of learning curves and multiple descent structures](https://mlsanity.com/paper/2212.06757/) (2022)
- [High-dimensional ridge regression with random features for non-identically distributed data with a variance profile](https://mlsanity.com/paper/2504.03035/) (2025)
- [auto-fpt: Automating Free Probability Theory Calculations for Machine Learning Theory](https://mlsanity.com/paper/2504.10754/) (2025)
- [An Asymptotic Theory of Chain-of-Thought in In-Context Learning](https://mlsanity.com/paper/2606.03217/) (2026)

## Abstract (arXiv)

We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type II1 von Neumann algebras, with a precise control of the domains of the rational expressions. The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists.