From Noncommutative Geometry to Random Matrix Theory

TL;DR

This paper reviews recent advances in the analytic study of random matrix models inspired by noncommutative geometry, focusing on spectral properties, phase transitions, and connections to quantum gravity.

Contribution

It introduces new random matrix ensembles based on noncommutative geometry and explores their spectral phase transitions and potential links to quantum gravity.

Findings

Ensembles exhibit spectral phase transitions.

Near phase transitions, they show manifold-like behavior.

Possible recovery of Liouville quantum gravity in certain limits.

Abstract

We review recent progress in the analytic study of random matrix models suggested by noncommutative geometry. One considers fuzzy spectral triples where the space of possible Dirac operators is assigned a probability distribution. These ensembles of Dirac operators are constructed as toy models of Euclidean quantum gravity on finite noncommutative spaces and display many interesting properties. The ensembles exhibit spectral phase transitions, and near these phase transitions they show manifold-like behavior. In certain cases one can recover Liouville quantum gravity in the double scaling limit. We highlight examples where bootstrap techniques, Coulomb gas methods, and Topological Recursion are applicable.