Research

In 2015, motivated by the Spectral Action Principle in Noncommutative Geometry, John Barrett proposed models of Euclidean quantum gravity that integrate over collections of finite matrix spectral triples called fuzzy spectral triples or fuzzy geometries. Such models are multi-tracial multi-matrix Hermitian matrix integrals. My works focus on studying such models analytically using techniques involving: Riemann-Hilbert problems, map enumeration, and the Schwinger-Dyson equations.

My thesis: Random Noncommutative Geometries, Map Enumeration, and Schwinger-Dyson Equations. 

1. Khalkhali, M. and Pagliaroli, N., 2020. Phase transition in random noncommutative geometries. Journal of Physics A: Mathematical and Theoretical, 54(3), p.035202. 

2. Khalkhali, M. and Pagliaroli, N., 2022. Spectral statistics of Dirac ensembles. Journal of Mathematical Physics, 63(5). 

3. Hessam, H., Khalkhali, M., Pagliaroli, N. and Verhoeven, L.S., 2022. From noncommutative geometry to random matrix theory. Journal of Physics A: Mathematical and Theoretical, 55(41), p.413002. 

4. Hessam, H., Khalkhali, M. and Pagliaroli, N., 2023. Double scaling limits of Dirac ensembles and Liouville quantum gravity. Journal of Physics A: Mathematical and Theoretical, 56(22), p.225201. 

5. Khalkhali, M. and Pagliaroli, N., 2024. Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry. Journal of High Energy Physics, 2024(5), pp.1-28. 

6. Khalkhali, M., Pagliaroli, N. and Verhoeven, L.S., 2025. Large N limit of fuzzy geometries coupled to fermions. Journal of Mathematical Physics, 66(5). 

7. Gamble, J., Khalkhali, M. and Pagliaroli, N., 2026. The Schwinger-Dyson equations for random fuzzy geometries coupled to matter. arXiv:2606.01343.

Given a "well-behaved" random matrix, by combining the Schwinger-Dyson equations and the positivity constraints of the Hamburger moment problem, one has a non-linear optimization problem. Solving truncated versions of this problem provides successive approximation of moments, critical points, and the distribution of eigenvalues of the random matrix. Our most recent work has extented this framework to random tensor models.

1. Hessam, H., Khalkhali, M. and Pagliaroli, N., 2022. Bootstrapping Dirac ensembles. Journal of Physics A: Mathematical and Theoretical, 55(33), p.335204. 

2. Khalkhali, M., Pagliaroli, N., Parfeni, A. and Smith, B., 2025. Bootstrapping the critical behavior of multi-matrix models. Journal of High Energy Physics, 2025(2), pp.1-35. 

3. Khalkhali, M. and Pagliaroli, N., 2025. Bootstrapping Noncommutative Geometry with Dirac Ensembles. arXiv:2512.08694. To appear as Chapter 5 of Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time. European Mathematical Society.

4. Pagliaroli, N, Pérez-Sánchez, Carlos I., and Smith, B., 2026. Bootstrapping random tensor integrals. arXiv:2604.19714.

1. Riddell, J., Pagliaroli, N.J. and Alhambra, Á.M., 2023. Concentration of quantum equilibration and an estimate of the recurrence time. SciPost Physics, 15(4), p.165. 

2. Riddell, J. and Pagliaroli, N., 2024. No-resonance conditions, random matrices, and quantum chaotic models. Journal of Statistical Physics, 191(11), p.141. 

3. Pollock, K., Kroth, J.D., Pagliaroli, N., Iadecola, T. and Riddell, J., 2025. Energy dynamics in a class of local random matrix Hamiltonians. Physical Review Research, 7(3), p.033129. 

1. Pagliaroli, N., 2025. Enumerating planar stuffed maps as hypertrees of mobiles. arXiv:2506.06086.