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 thesis work focused 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.
I am also interested in studying matrix and tensor models usign the method of Bootstrapping with psoitvity. 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 extended this framework to random tensor models.
I also work on applications of random matrices in quantum statistical mechanics and quantum information. An upcoming article studies the entropy of random subspaces of bipartite quantum systems using Weingarten Calculus and Free Probability Theory.
Gamble, J., Khalkhali, M. and Pagliaroli, N., 2026. The Schwinger-Dyson equations for random fuzzy geometries coupled to matter. arXiv:2606.01343.
Pagliaroli, N, Pérez-Sánchez, Carlos I., and Smith, B., 2026. Bootstrapping random tensor integrals. arXiv:2604.19714.
Khalkhali, M. and Pagliaroli, N., 2026. Bootstrapping Noncommutative Geometry with Dirac Ensembles. arXiv:2512.08694. Chapter 5 of Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time. European Mathematical Society.
Khalkhali, M., Pagliaroli, N., Parfeni, A. and Smith, B., 2025. Bootstrapping the critical behavior of multi-matrix models. arXiv:2409.07565. Journal of High Energy Physics, 2025(2), pp.1-35.
Pollock, K., Kroth, J.D., Pagliaroli, N., Iadecola, T. and Riddell, J., 2025. Energy dynamics in a class of local random matrix Hamiltonians. arXiv:2502.05045. Physical Review Research, 7(3), p.033129.
Pagliaroli, N., 2025. Enumerating planar stuffed maps as hypertrees of mobiles. arXiv:2506.06086.
Riddell, J. and Pagliaroli, N., 2024. No-resonance conditions, random matrices, and quantum chaotic models. arXiv:2307.05417. Journal of Statistical Physics, 191(11), p.141.
Khalkhali, M. and Pagliaroli, N., 2024. Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry. arXiv:2312.10530. Journal of High Energy Physics, 2024(5), pp.1-28.
Khalkhali, M., Pagliaroli, N. and Verhoeven, L.S., 2025. Large N limit of fuzzy geometries coupled to fermions. arXiv:405.05056. Journal of Mathematical Physics, 66(5).
Hessam, H., Khalkhali, M. and Pagliaroli, N., 2023. Double scaling limits of Dirac ensembles and Liouville quantum gravity. arXiv:2204.14206. Journal of Physics A: Mathematical and Theoretical, 56(22), p.225201.
Riddell, J., Pagliaroli, N.J. and Alhambra, Á.M., 2023. Concentration of quantum equilibration and an estimate of the recurrence time. arXiv:2206.07541. SciPost Physics, 15(4), p.165.
Hessam, H., Khalkhali, M., Pagliaroli, N. and Verhoeven, L.S., 2022. From noncommutative geometry to random matrix theory. arXiv:2204.14216. Journal of Physics A: Mathematical and Theoretical, 55(41), p.413002.
Hessam, H., Khalkhali, M. and Pagliaroli, N., 2022. Bootstrapping Dirac ensembles. arXiv:2107.10333. Journal of Physics A: Mathematical and Theoretical, 55(33), p.335204.
Khalkhali, M. and Pagliaroli, N., 2022. Spectral statistics of Dirac ensembles. arXiv:2109.12741. Journal of Mathematical Physics, 63(5).
Khalkhali, M. and Pagliaroli, N., 2020. Phase transition in random noncommutative geometries. arXiv:2006.02891. Journal of Physics A: Mathematical and Theoretical, 54(3), p.035202.