Monte Carlo simulations of random non-commutative geometries

Abstract

Random non-commutative geometries are introduced by integrating over the space of Dirac operators that form a spectral triple with a fixed algebra and Hilbert space. The cases with the simplest types of Clifford algebra are investigated using Monte Carlo simulations to compute the integrals. Various qualitatively different types of behaviour of these random Dirac operators are exhibited. Some features are explained in terms of the theory of random matrices but other phenomena remain mysterious. Some of the models with a quartic action of symmetry-breaking type display a phase transition. Close to the phase transition the spectrum of a typical Dirac operator shows manifold-like behaviour for the eigenvalues below a cut-off scale.

35 pages, author manuscript accepted by journal

References in corpus (4)

• Nonperturbative Quantum Gravity
• A Lorentzian version of the non-commutative geometry of the standard model of particle physics
• Quantum Gravity on the Lattice
• Matrix geometries and fuzzy spaces as finite spectral triples

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• Spectral Statistics of Dirac Ensembles
• From Noncommutative Geometry to Random Matrix Theory
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• Surgery in colored tensor models
• One-loop corrections to the spectral action
• Double scaling limits of Dirac ensembles and Liouville quantum gravity
• BV quantization of dynamical fuzzy spectral triples
• Renormalizing Yukawa interactions in the standard model with matrices and noncommutative geometry
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• Computational explorations of a deformed fuzzy sphere
• A ribbon graph derivation of the algebra of functional renormalization for random multi-matrices with multi-trace interactions
• The loop equations for noncommutative geometries on quivers
• Pinpointing Triple Point of Noncommutative Matrix Model with Curvature
• Eigenvalue-flipping Algorithm for Matrix Monte Carlo