Constructive Matrix Theory for Higher Order Interaction

Krajewski, Thomas; Rivasseau, Vincent; Sazonov, Vasily

doi:10.48550/arxiv.1712.05670

Cited by 3 publications

(6 citation statements)

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“…Then, using (11), each term in the double sum (10) can be rewritten as a moment of a Gaussian distribution, i.e. the convergent series is expressed as a re-summation of the standard perturbation theory…”

Section: Cs Methods and Dimensional Regularizationmentioning

confidence: 99%

Infinite lattice models by an expansion with a non-Gaussian initial approximation

Ivanov

Sazonov

2019

Physics Letters B

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Recently, a convergent series employing a non-Gaussian initial approximation was constructed and shown to be an effective computational tool for the finite size lattice models with a polynomial interaction. Here we show that the Borel summability is a sufficient condition for the correctness of the convergent series applied to infinite lattice models. We test the numerical workability of the convergent series method by examine one-and two-dimensional φ 4 -infinite lattice models. The comparison of the convergent series computations and the infinite lattice extrapolations of the Monte Carlo simulations reveals an agreement between two approaches.

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Section: Cs Methods and Dimensional Regularizationmentioning

confidence: 99%

Infinite lattice models by an expansion with a non-Gaussian initial approximation

Ivanov

Sazonov

2019

Physics Letters B

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“…The Fuss-Catalan function T p (u) is well known in the literature. Here we follow the presentation in [25]. T p (u) is defined by the implicit equation:…”

Section: The Tensor Gaussian Orthogonal Ensemblementioning

confidence: 99%

On the generalization of the Wigner semicircle law to real symmetric tensors

Gurau

2020

Preprint

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We propose a simple generalization of the matrix resolvent to a resolvent for real symmetric tensors T ∈ ⊗ p R N of order p ≥ 3. The tensor resolvent yields an integral representation for a class of tensor invariants and its singular locus can be understood in terms of the real eigenvalues of tensors.We then consider a random Gaussian (real symmetric) tensor. We show that in the large N limit the expected resolvent has a finite cut in the complex plane and that the associated "spectral density", that is the discontinuity at the cut, obeys a universal law which generalizes the Wigner semicircle law to arbitrary order.Finally, we consider a spiked tensor for p ≥ 3, that is the sum of a fixed tensor b v ⊗p with v ∈ R N (the signal) and a random Gaussian tensor T (the noise). We show that in the large N limit the expected resolvent undergoes a sharp transition at some threshold value of the signal to noise ratio b which we compute analytically.

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“…From now on we consider the free part of the action to be: We introduce an ultraviolet cutoff Λ and an infrared one k and we replace C by C Λ k , the covariance with cutoffs 19 . In order to simplify the notation we sometimes suppress the UV cutoff, but one should remember that for now the UV cutoff is present 20 We parametrize the renormalization group flow by the effective action at scale k:…”

Section: B the Renormalization (Semi-)groupmentioning

confidence: 99%

“…This strategy brings mixed results. With the exception of some models in dimension zero 1 [18][19][20] step 3 is almost never considered. Step 2 is again considered mostly in dimension zero 1 By dimension we mean the dimension of the space or space-time on which the vector, matrix or tensor field theory is defined, that is QCD is a theory in 4 dimensions [15].…”

Section: Introductionmentioning

confidence: 99%

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Notes on Tensor Models and Tensor Field Theories

Gurău¹

2019

Preprint

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Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the 1/N expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.

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Constructive Matrix Theory for Higher Order Interaction

Cited by 3 publications

References 0 publications

Infinite lattice models by an expansion with a non-Gaussian initial approximation

Infinite lattice models by an expansion with a non-Gaussian initial approximation

On the generalization of the Wigner semicircle law to real symmetric tensors

Notes on Tensor Models and Tensor Field Theories

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