Spectral statistics of Dirac ensembles

Khalkhali, Masoud; Nathan, Pagliaroli,

doi:10.1063/5.0078267

Cited by 9 publications

(22 citation statements)

References 14 publications

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“…These objects are interesting purely from a random matrix perspective, of which very little is known in general. Relatively recently some universal properties were established in [30].…”

Section: Dirac Ensemblesmentioning

confidence: 99%

“…As one might guess, there is a deep and not well understood relationship between them. The spectrum of the Dirac operator itself is not fully understood but displays some universal behaviors as seen in [30] and spectral phase transitions as seen in [5,24,29]. We are interested only in the simplest cases for now, partly because of the lack of analytical tools needed to study multi-matrix models.…”

Section: Dirac Ensemblesmentioning

confidence: 99%

“…We define SM g k (v) as the set of all unstable stuffed maps of genus g, with v vertices and k boundaries. It was proven in [30] that this set is finite, allowing us to define the following formal series:…”

Section: Stuffed Mapsmentioning

confidence: 99%

“…Taking the large N limit of matrix models is equivalent to counting various types of planar maps [7]. As proven in [30], Dirac ensembles of any dimension have a genus expansion. In particular, the leading order term, found by a large N limit of appropriately scaled quantities, amounts to counting strictly planar maps.…”

Section: Introductionmentioning

confidence: 99%

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Double scaling limits of Dirac ensembles and Liouville quantum gravity

Hessam

Khalkhali

Nathan

2023

J. Phys. A: Math. Theor.

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In this paper we study ensembles of finite real spectral triples equipped with a path integralover the space of possible Dirac operators. In the noncommutative geometric setting of spectraltriples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus,this path integral serves as a noncommutative analogue of integration over metrics, a key featureof a theory of quantum gravity. From these integrals in the so-called double scaling limit we derivecritical exponents of minimal models from Liouville conformal field theory coupled with gravity.Additionally, the asymptotics of the partition function of these models satisfy differential equationssuch as Painlev´e I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory.This is all proven using well-established and rigorous techniques from random matrix theory.

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“…These objects are interesting purely from a random matrix perspective, of which very little is known in general. Relatively recently some universal properties were established in [30].…”

Section: Dirac Ensemblesmentioning

confidence: 99%

Section: Dirac Ensemblesmentioning

confidence: 99%

Section: Stuffed Mapsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Hessam

Khalkhali

Nathan

2023

J. Phys. A: Math. Theor.

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“…'Random NCG' is a short name for the construction of probability measures on families of finite-dimensional 2 Dirac operators, which is in line with equation (1.1). Since this partition function resembles the canonical ensemble in statistical physics, 'random NCG' is called 'Dirac ensembles' elsewhere [49]. Also the terminology 'dynamical fuzzy spectral triples' is used in [35], where the Batalin-Vilkovisky formalism is addressed for this type of models.…”

Section: Recent and Parallel Progress In Random Ncgmentioning

confidence: 99%

Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

Perez-Sanchez

2022

J. Noncommut. Geom.

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A fuzzy geometry is a certain type of spectral triple whose Dirac operator crucially turns out to be a finite matrix. This notion incorporates familiar examples like fuzzy spheres and fuzzy tori. In the framework of random noncommutative geometry, we use Barrett's characterization of Dirac operators of fuzzy geometries in order to systematically compute the spectral action S.D/ D Tr f .D/ for 2n-dimensional fuzzy geometries. In contrast to the original Chamseddine-Connes spectral action, we take a polynomial f with f .x/ ! 1 as jxj ! 1 in order to obtain a well-defined path integral that can be stated as a random matrix model with action of the type S.D/ D N tr F C P i tr A i tr B i , being F , A i and B i noncommutative polynomials in 2 2n 1 complex N N matrices that parametrize the Dirac operator D. For arbitrary signature-thus for any admissible KO-dimension-formulas for 2-dimensional fuzzy geometries are given up to a sextic polynomial, and up to a quartic polynomial for 4-dimensional ones, with focus on the octo-matrix models for Lorentzian and Riemannian signatures. The noncommutative polynomials F , A i and B i are obtained via chord diagrams and satisfy: independence of N ; self-adjointness of the main polynomial F (modulo cyclic reordering of each monomial); also up to cyclicity, either self-adjointness or anti-self-adjointness of A i and B i simultaneously, for fixed i. Collectively, this favors a free probabilistic perspective for the large-N limit we elaborate on.

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Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

Khalkhali,

Pagliaroli

2024

J. High Energ. Phys.

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We compute the first twenty moments of three convergent quartic bi-tracial 2-matrix ensembles in the large N limit. These ensembles are toy models for Euclidean quantum gravity originally proposed by John Barrett and collaborators. A perturbative solution is found for the first twenty moments using the Schwinger-Dyson equations and properties of certain bi-colored unstable maps associated to the model. We then apply a result of Guionnet et al. to show that the perturbative and convergent solution coincide for a small neighbourhood of the coupling constants. For each model we compute an explicit expression for the free energy, critical points, and critical exponents in the large N limit. In particular, the string susceptibility is found to be γ = 1/2, hinting that the associated universality class of the model is the continuous random tree.

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Spectral statistics of Dirac ensembles

Cited by 9 publications

References 14 publications

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Double scaling limits of Dirac ensembles and Liouville quantum gravity

Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models

Coloured combinatorial maps and quartic bi-tracial 2-matrix ensembles from noncommutative geometry

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