# Persistence diagrams of random matrices via Morse theory: universality and a new spectral diagnostic

arXiv: https://arxiv.org/abs/2603.27903 · HTML: https://mlsanity.com/paper/2603.27903/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 19 references that are in the map.
- First posted on arXiv 29 Mar 2026.

## Builds on (most cited first, 1 of 1)

- [On the choice of weight functions for linear representations of persistence diagrams](https://mlsanity.com/paper/1807.03678/) (2018)

## Abstract (arXiv)

We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = xT M x restricted to the unit sphere Sn-1 is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing sk = λk+1 - λk. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n-0.6. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio langle r rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which langle r rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.