A random matrix model with non-pairwise contracted indices

Lionni, Luca; Sasakura, Naoki

doi:10.1093/ptep/ptz057

Cited by 17 publications

(59 citation statements)

References 40 publications

(154 reference statements)

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“…In this paper, however, we will take a different strategy. This is because the new strategy makes more transparent the rather complicated counting of combinatorics performed in [21], and make it straightforward to include the extra coupling λ d U d (φ) and also to consider the next order in t. For λ d = 0, the new strategy gives essentially the same result as [21] in the leading order, as commented below (17).…”

Section: Observablesmentioning

confidence: 86%

“…The f N,R (t) has also the property that it is a decreasing positive function of t with f N,R (0) = 1 for real t. This property provides a good criterion for assessing the validity of an approximation of f N,R (t). In the previous paper [21], f N,R (t) in the leading order of 1/R has been determined by a Feynman diagrammatic method with the result,…”

Section: The Modelmentioning

confidence: 99%

“…In this paper, we numerically study the matrix model by the Monte Carlo simulation with the standard Metropolis update method. This contrasts with the perturbative analytical computations performed in the previous paper [21]. We also perform some additional analytical computations to compare with the numerical results.…”

Section: Introductionmentioning

confidence: 97%

“…Though this toy wave function is simpler than the actual one, it is still difficult to perform thorough analyses, because the dimension of the argument (a symmetric tensor with three indices) of the wave function is very large with the order of ∼ N 3 /6. Therefore, as an additional simplification, the authors in [21] considered a model that can be obtained by integrating over the argument of the toy wave function. This gives a dynamical system of a matrix, say φ i a (a = 1, 2, .…”

Section: Introductionmentioning

confidence: 99%

“…• The expectation values of some observables are analytically computed in the leading order, mostly based on the treatment in the previous paper [21], and are compared with the numerical results. Good agreement between them is obtained outside the vicinity of the transition region, while there exist deviations in the transition region.…”

Section: Introductionmentioning

confidence: 99%

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Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

Sasakura

Takeuchi

2020

Eur. Phys. J. C

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We study a matrix model that has φ i a (a = 1, 2, . . . , N, i = 1, 2, . . . , R) as its dynamical variable, whose lower indices are pairwise contracted, but upper ones are not always done so. This matrix model has a motivation from a tensor model for quantum gravity, and is also related to the physics of glasses, because it has the same form as what appears in the replica trick of the spherical p-spin model for spin glasses, though the parameter range of our interest is different. To study the dynamics, which in general depends on N and R, we perform Monte Carlo simulations and compare with some analytical computations in the leading and the next-leading orders. A transition region has been found around R ∼ N 2 /2, which matches a relation required by the consistency of the tensor model. The simulation and the analytical computations agree well outside the transition region, but not in this region, implying that some relevant configurations are not properly included by the analytical computations. With a motivation coming from the tensor model, we also study the persistent homology of the configurations generated in the simulations, and have observed its gradual change from S 1 to higher dimensional cycles with the increase of R around the transition region.

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Section: Observablesmentioning

confidence: 86%

Section: The Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

Sasakura

Takeuchi

2020

Eur. Phys. J. C

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The tensor of the exact circle: reconstructing geometry

Obster

2023

Phys. Scr.

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Developing a theory for quantum gravity is one of the big open questions in theoretical high-energy physics. Recently, a tensor model approach has been considered that treats tensors as the generators of commutative non-associative algebras, which might be an appropriate interpretation of the canonical tensor model. In this approach, the non-associative algebra is assumed to be a low-energy description of the so-called associative closure, which gives the full description of spacetime including the high-energy modes. In the previous work it has been shown how to (re)construct a topological space with a measure on it, and one of the prominent examples that was used to develop the framework was the exact circle. In this work we will further investigate this example, and show that it is possible to reconstruct the full Riemannian geometry by reconstructing the metric tensor. Furthermore, it is demonstrated how diﬀeomorphisms behave in this formalism, ﬁrstly by considering a speciﬁc class of diﬀeomorphisms of the circle, namely the ellipses, and subsequently by performing an explicit diﬀeomorphism to “smoothen” sets of points generated by the tensor rank decomposition.

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Phases of a matrix model with non-pairwise index contractions

Obster

Sasakura

2020

Progress of Theoretical and Experimental Physics

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Abstract Recently a matrix model with non-pairwise index contractions has been studied in the context of the canonical tensor model, a tensor model for quantum gravity in the canonical formalism. This matrix model also appears in the same form with different ranges of parameters and variables, when the replica trick is applied to the spherical $p$-spin model ($p=3$) in spin glass theory. Previous studies of this matrix model suggested the presence of a continuous phase transition around $R\sim N^2/2$, where $N$ and $R$ designate its matrix size $N\times R$. This relation between $N$ and $R$ intriguingly agrees with a consistency condition of the tensor model in the leading order of $N$, suggesting that the tensor model is located near or on the continuous phase transition point and therefore its continuum limit is automatically taken in the $N\rightarrow \infty$ limit. In the previous work, however, the evidence for the phase transition was not satisfactory due to the slowdown of the Monte Carlo simulations. In this work, we provide a new setup for Monte Carlo simulations by integrating out the radial direction of the matrix. This new strategy considerably improves the efficiency, and allows us to clearly show the existence of the phase transition. We also present various characteristics of the phases, such as dynamically generated dimensions of configurations, cascade symmetry breaking and a parameter zero limit, and discuss their implications for the canonical tensor model.

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A random matrix model with non-pairwise contracted indices

Cited by 17 publications

References 40 publications

Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

The tensor of the exact circle: reconstructing geometry

Phases of a matrix model with non-pairwise index contractions

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