Monte Carlo simulations of random non-commutative geometries

Barrett, John W.; Glaser, Lisa

doi:10.1088/1751-8113/49/24/245001

Cited by 36 publications

(149 citation statements)

References 28 publications

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“…After testing the formalism on spheres and tori, the distances between the average spectra of some random geometries are computed. Also, the distance to the fuzzy sphere reveals that the average geometries are indeed close to the fuzzy sphere near the phase transition, as first suggested in [10]. The conclusions of the paper are presented in section V.…”

Section: Introductionsupporting

confidence: 62%

“…For particular values of g 4 and g 2 it can be derived from the lowest order expansion of the heat kernel. With g 4 = 1 and varying g 2 < 0, the random fuzzy spaces show a phase transition, which was described in more detail in [10,18]. The location of the phase transition depends on the Clifford module type (p, q).…”

Section: Fuzzy Spacesmentioning

confidence: 95%

“…For other values of N the behaviour is more complicated, so these cases are not explored here. The last class of fuzzy spaces examined is the class of random fuzzy spaces defined in [10]. The Dirac operator has an explicit expression in [15] using several N × N Hermitian matrices H i and anti-Hermitian traceless matrices L j .…”

Section: Fuzzy Spacesmentioning

confidence: 99%

“…This allows an implementation of a Monte Carlo algorithm on these geometries. Recent work using a simple action showed that this approach can give rise to a phase transition, close to which indications of 2-dimensional manifold-like behaviour were observed [10]. This was done by comparing the spectral density of the random fuzzy space to the spectral density of the fuzzy sphere by eye, but clearly quantitative tools are necessary.…”

Section: Introductionmentioning

confidence: 99%

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Spectral estimators for finite non-commutative geometries

Barrett¹,

Druce²,

Glaser³

2019

J. Phys. A: Math. Theor.

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A finite non-commutative geometry consists of a fuzzy space together with a Dirac operator satisfying the axioms of a real spectral triple. This paper addresses the question of how to extract information about these geometries from the spectrum of the Dirac operator. Since the Dirac operator is a finite-dimensional matrix, the usual asymptotics of the eigenvalues makes no sense and is replaced by measurements of the spectrum at a finite energy scale. The spectral dimension of the square of the Dirac operator is improved to provide a new spectral measure of the dimension of a space called the spectral variance. Similarly, the volume of a space can be computed from the spectrum once the dimension is known. Two methods of doing this are investigated: the well-known Dixmier trace and a recent improvement due to Abel Stern. Finally, the distance between two geometries is investigated by comparing the spectral zeta functions using the method of Cornelissen and Kontogeorgis. All of these techniques are tested on the explicit examples of the fuzzy spheres and fuzzy tori, which can be regarded as approximations of the usual Riemannian sphere and flat tori. Then they are applied to characterise some random fuzzy spaces using data generated by a Monte Carlo simulation.

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Section: Introductionsupporting

confidence: 62%

Section: Fuzzy Spacesmentioning

confidence: 95%

Section: Fuzzy Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral estimators for finite non-commutative geometries

Barrett¹,

Druce²,

Glaser³

2019

J. Phys. A: Math. Theor.

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“…with S a trace of powers of D, over finite-rank Dirac operators, as a possible nonperturbative description for quantum gravitational phenomena [15,16,17].…”

Section: Background: Noncommutative Geometry and The Cutoff Scalementioning

confidence: 99%

Understanding truncated non-commutative geometries through computer simulations

Glaser

Stern

2020

Journal of Mathematical Physics

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When aiming to apply mathematical results of non-commutative geometry to physical problems the question arises how they translate to a context in which only a part of the spectrum is known.In this article we aim to detect when a finite-dimensional triple is the truncation of the Dirac spectral triple of a spin manifold. To that end, we numerically investigate the restriction that the higher Heisenberg equation [1] places on a truncated Dirac operator. We find a bounded perturbation of the Dirac operator on the Riemann sphere that induces the same Chern class. arXiv:1909.08054v1 [math-ph]

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Eigenvalue-flipping algorithm for matrix Monte Carlo

Kováčik

Tekel

2022

J. High Energ. Phys.

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Many physical systems can be described in terms of matrix models that we often cannot solve analytically. Fortunately, they can be studied numerically in a straightforward way. Many commonly used algorithms follow the Monte Carlo method, which is efficient for small matrix sizes but cannot guarantee ergodicity when working with large ones. In this paper, we propose an improvement of the algorithm that, for a large class of matrix models, allows to tunnel between various vacua in a proficient way, where sign change of eigenvalues is proposed externally. We test the method on two models: the pure potential matrix model and the scalar field theory on the fuzzy sphere.

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Monte Carlo simulations of random non-commutative geometries

Cited by 36 publications

References 28 publications

Spectral estimators for finite non-commutative geometries

Spectral estimators for finite non-commutative geometries

Understanding truncated non-commutative geometries through computer simulations

Eigenvalue-flipping algorithm for matrix Monte Carlo

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