From random matrices to random tensors

2021 ◽  
pp. 166-177
Author(s):  
Adrian Tanasa
Keyword(s):  
Matrix Models ◽  
Random Matrices ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Random Surface ◽  
The Third ◽  
Planar Surfaces ◽  
Tensor Models ◽  
Feynman Graphs ◽  
The Matrix

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

scholarly journals Asymptotic Expansion of the Multi-Orientable Random Tensor Model

10.37236/4629 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Eric Fusy ◽  
Adrian Tanasa
Keyword(s):  
Asymptotic Expansion ◽  
Matrix Models ◽  
Three Dimensional ◽  
Natural Generalization ◽  
General Term ◽  
Tensor Model ◽  
Half Integer ◽  
Tensor Models ◽  
Random Tensor

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expanion in $N$, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.

scholarly journals Renormalization Group Approach to the Continuum Limit of Matrix Models of Quantum Gravity With Preferred Foliation

2021 ◽  
Vol 9 ◽  
Author(s):  
Alicia Castro ◽  
Tim Andreas Koslowski
Keyword(s):  
Renormalization Group ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Fixed Points ◽  
Renormalization Group Equation ◽  
Two Dimensions ◽  
Gravity Models ◽  
Group Equation ◽  
Tensor Models ◽  
The Matrix

This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson [1], which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions.

Alternaria toxins and their effects on host plants

1995 ◽  
Vol 73 (S1) ◽  
pp. 453-458 ◽  
Author(s):  
Hiroshi Otani ◽  
Keisuke Kohmoto ◽  
Motoichiro Kodama
Keyword(s):  
Plasma Membrane ◽  
Host Plants ◽  
Plasma Membranes ◽  
Early Effect ◽  
The Third ◽  
The Matrix ◽  
Electrolyte Loss ◽  
Host Specific Toxins ◽  
Acid Structure ◽  
Malate Oxidation

There are now nine or more Alternaria pathogens that produce host-specific toxins, and the structures of most of the toxins have been elucidated. Alternaria host-specific toxins are classified in three groups in terms of the primary site action. ACT-, AF-, and AK-toxins have in common an epoxy-decatrienoic acid structure and exert their primary effect on the plasma membrane of susceptible cells. A rapid increase in electrolyte loss from tissues and invaginations in the plasma membranes are common effects of these toxins. The second group is represented by ACR(L)-toxin, which induces changes in mitochondria, including swelling, vesiculation of cristae, decrease in the electron density of the matrix, increase in the rate of NADH oxidation, and inhibition of malate oxidation. The third group consists of AM-toxin, which appears to exert an early effect on both chloroplasts and plasma membranes. AM-toxin induces vesiculation of grana lamellae, inhibition of CO2 fixation, invagination of plasma membranes, and electrolyte loss. The roles of host-specific toxins in pathogenesis are discussed. Key words: Alternaria, host-specific toxin, plasma membrane, mitochondrion, chloroplast.

scholarly journals HILL CIPHER ON SHAMIR’S THREE PASS PROTOCOL

2017 ◽  
Author(s):  
hasdiana
Keyword(s):  
Matrix Element ◽  
Execution Time ◽  
Invertible Matrix ◽  
Multidisciplinary Research ◽  
The Third ◽  
Xor Operation ◽  
Hill Cipher ◽  
The Matrix ◽  
Level Of Difficulty ◽  
Time Required

This preprint has been presented in the 3rd International Conference on Multidisciplinary Research, Medan, october 16 – 18, 2014---In this study the authors use the scheme of Shamir's Three Pass Protocol for Hill Cipher operation. Scheme of Shamir's Three Pass Protocol is an attractive scheme that allows senders and receivers to communicate without the key exchange. Hill Cipher is chosen because of the key-shaped matrix, which is expected to complicate the various techniques of cryptanalyst. The results of this study indicate that the weakness of the scheme of Shamir's Three Pass Protocol for XOR operation is not fully valid if it is used for Hill Cipher operations. Cryptanalyst can utilize only the third ciphertext that invertible. Matrix transpose techniques in the ciphertext aims to difficulties in solving this algorithm. The original ciphertext generated in each process is different from the transmitted ciphertext. The level of difficulty increases due to the use of larger key matrix. The amount of time required for the execution of the program depends on the length of the plaintext and the value of the matrix element. Plaintext has the same length produce different execution time depending on the value of the key elements of the matrix used.

scholarly journals An asymptotic property of large matrices with identically distributed Boolean independent entries

2019 ◽  
Vol 22 (04) ◽  
pp. 1950024
Author(s):  
Mihai Popa ◽  
Zhiwei Hao
Keyword(s):  
Recent Result ◽  
Recent Work ◽  
Random Matrices ◽  
Asymptotic Property ◽  
Limit Distribution ◽  
Asymptotic Independence ◽  
Moment Conditions ◽  
The Matrix ◽  
Combinatorial Condition ◽  
Independence Relations

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically [Formula: see text]-diagonal and Boolean independent from each other. This paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that the random matrices considered are asymptotically Boolean independent from some of their partial transposes. The main results of the paper are based on combinatorial techniques.

scholarly journals Edge universality for non-Hermitian random matrices

2020 ◽  
Author(s):  
Giorgio Cipolloni ◽  
László Erdős ◽  
Dominik Schröder
Keyword(s):  
Unit Circle ◽  
Random Matrices ◽  
Matrix Elements ◽  
Spectral Edge ◽  
Ginibre Ensemble ◽  
Eigenvalue Statistics ◽  
The Matrix ◽  
Independent Identically Distributed

Abstract We consider large non-Hermitian real or complex random matrices $$X$$ X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of $$X$$ X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble.

scholarly journals Bernstein-type inequality for a class of dependent random matrices

2016 ◽  
Vol 05 (02) ◽  
pp. 1650006 ◽  
Author(s):  
Marwa Banna ◽  
Florence Merlevède ◽  
Pierre Youssef
Keyword(s):  
Random Matrices ◽  
Type Inequality ◽  
Bernstein Inequality ◽  
Mathematical Statistics ◽  
Strong Mixing ◽  
Mixing Conditions ◽  
Bernstein Type ◽  
Bernstein Type Inequality ◽  
The Matrix ◽  
The Laplace Transform

In this paper, we obtain a Bernstein-type inequality for the sum of self-adjoint centered and geometrically absolutely regular random matrices with bounded largest eigenvalue. This inequality can be viewed as an extension to the matrix setting of the Bernstein-type inequality obtained by Merlevède et al. [Bernstein inequality and moderate deviations under strong mixing conditions, in High Dimensional Probability V: The Luminy Volume, Institute of Mathematical Statistics Collection, Vol. 5 (Institute of Mathematical Statistics, Beachwood, OH, 2009), pp. 273–292.] in the context of real-valued bounded random variables that are geometrically absolutely regular. The proofs rely on decoupling the Laplace transform of a sum on a Cantor-like set of random matrices.

scholarly journals ON FINITE RANK DEFORMATIONS OF WIGNER MATRICES II: DELOCALIZED PERTURBATIONS

2013 ◽  
Vol 02 (01) ◽  
pp. 1250015 ◽  
Author(s):  
DAVID RENFREW ◽  
ALEXANDER SOSHNIKOV
Keyword(s):  
Random Matrices ◽  
Finite Rank ◽  
Fourth Moment ◽  
Wigner Matrices ◽  
Wigner Random Matrices ◽  
The Matrix

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and Féral for perturbations whose eigenvectors are delocalized.

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