From Noncommutative Geometry to Random Matrix Theory

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.

arXiv comment40 pages, 16 figures. Accepted to appear as a topical review in Journal of Physics A: Mathematical and Theoretical. The new version is largely expanded with new calculations and additional topics

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Cited by in corpus (5)

• Random Finite Noncommutative Geometries and Topological Recursion
• Double scaling limits of Dirac ensembles and Liouville quantum gravity
• BV quantization of dynamical fuzzy spectral triples
• Pinpointing Triple Point of Noncommutative Matrix Model with Curvature
• The loop equations for noncommutative geometries on quivers