Random tensor models—the U(N)D-invariant model

2021 ◽  
pp. 178-208
Author(s):  
Adrian Tanasa
Keyword(s):  
Scaling Limit ◽  
Point Function ◽  
Invariant Model ◽  
Invariant Tensor ◽  
The Third ◽  
Tensor Models ◽  
Random Tensor ◽  
Combinatorial Methods ◽  
Double Scaling Limit ◽  
The Way

In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.

scholarly journals Double scaling limit for the O(N)3-invariant tensor model

2022 ◽  
Author(s):  
Valentin Bonzom ◽  
Victor Nador ◽  
Adrian Tanasa
Keyword(s):  
Scaling Limit ◽  
Tensor Model ◽  
Invariant Tensor ◽  
Generating Series ◽  
Tensor Models ◽  
Feynman Graphs ◽  
Diagrammatic Analysis ◽  
Finite Sum ◽  
Double Scaling Limit ◽  
The Continuum

Abstract We study the double scaling limit of the O(N)3-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the 1/N expansion as a finite sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the 1/N expansion. This leads to a double scaling limit which picks up contributions from all orders in the 1/N expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of the Feynman graphs, as well as an analytic analysis of the singularities of the relevant generating series.

scholarly journals The double scaling limit of random tensor models

2014 ◽  
Vol 2014 (9) ◽  
Author(s):  
Valentin Bonzom ◽  
Razvan Gurau ◽  
James P. Ryan ◽  
Adrian Tanasa
Keyword(s):  
Scaling Limit ◽  
Tensor Models ◽  
Random Tensor ◽  
Double Scaling Limit

NONPERTURBATIVE 2D QUANTUM GRAVITY VIA SUPERSYMMETRIC STRING

1990 ◽  
Vol 05 (30) ◽  
pp. 2565-2572 ◽  
Author(s):  
MAREK KARLINER ◽  
SASHA MIGDAL
Keyword(s):  
Quantum Gravity ◽  
Ground State ◽  
Scaling Limit ◽  
Linear Ordinary Differential Equation ◽  
Third Order ◽  
Quantum Oscillations ◽  
The Third ◽  
Nonperturbative Solution ◽  
Distribution Of Eigenvalues ◽  
Double Scaling Limit

The Parisi-Marinari suggestion to treat 2d quantum gravity as ground state of the 1d supersymmetric string is elaborated in some detail. The third order linear ordinary differential equation describing in the double scaling limit the distribution of eigenvalues of the random matrix (i.e., the Liouville field) is derived and studied numerically. Unlike the Painlevé equation, our equation leads to continuous spectrum; however, the nonperturbative effects display themselves as quantum oscillations on top of smooth WKB distribution. Nonperturbative solution is free of any ambiguities.

scholarly journals Towards a double scaling limit for tensor models: probing sub-dominant orders

2014 ◽  
Vol 16 (6) ◽  
pp. 063048 ◽  
Author(s):  
Wojciech Kamiński ◽  
Daniele Oriti ◽  
James P Ryan
Keyword(s):  
Scaling Limit ◽  
Tensor Models ◽  
Double Scaling Limit

Random tensor models—the O(N)3-invariant model

2021 ◽  
pp. 234-259
Author(s):  
Adrian Tanasa
Keyword(s):  
Critical Exponents ◽  
Perturbative Expansion ◽  
Critical Regime ◽  
Leading Order ◽  
Invariant Model ◽  
Analytic Combinatorics ◽  
Tensor Models ◽  
Feynman Graphs ◽  
Random Tensor

We define in this chapter a class of tensor models endowed with O(N)3-invariance, N being again the size of the tensor. This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model and the U(N)3-invariant models treated in the previous two chapters. We first exhibit the existence of a large N expansion for such a model with general interactions (non-necessary quartic). We then focus on the quartic model and we identify the leading order and next-to-leading Feynman graphs of the large N expansion. Finally, we prove the existence of a critical regime and we compute the so-called critical exponents. This is achieved through the use of various analytic combinatorics techniques.

Géographie et cartographie fictionnelles dans l’Utopie (1516) de Thomas More

Moreana ◽  
2010 ◽  
Vol 47 (Number 181- (3-4) ◽  
pp. 9-68
Author(s):  
Jean Du Verger
Keyword(s):  
Complex Network ◽  
Narrative Strategies ◽  
Thomas More ◽  
The Third ◽  
Editio Princeps ◽  
Shed Light ◽  
The Way

The philosophical and political aspects of Utopia have often shadowed the geographical and cartographical dimension of More’s work. Thus, I will try to shed light on this aspect of the book in order to lay emphasis on the links fostered between knowledge and space during the Renaissance. I shall try to show how More’s opusculum aureum, which is fraught with cartographical references, reifies what Germain Marc’hadour terms a “fictional archipelago” (“The Catalan World Atlas” (c. 1375) by Abraham Cresques ; Zuane Pizzigano’s portolano chart (1423); Martin Benhaim’s globe (1492); Martin Waldseemüller’s Cosmographiae Introductio (1507); Claudius Ptolemy’s Geographia (1513) ; Benedetto Bordone’s Isolario (1528) ; Diogo Ribeiro’s world map (1529) ; the Grand Insulaire et Pilotage (c.1586) by André Thevet). I will, therefore, uncover the narrative strategies used by Thomas More in a text which lies on a complex network of geographical and cartographical references. Finally, I will examine the way in which the frontispiece of the editio princeps of 1516, as well as the frontispiece of the third edition published by Froben at Basle in 1518, clearly highlight the geographical and cartographical aspect of More’s narrative.

Ragam Qiraat Al-Qur'an

SUHUF ◽  
2015 ◽  
Vol 2 (1) ◽  
pp. 53-72
Author(s):  
Ahmad Fathoni
Keyword(s):  
The Third ◽  
The Difference ◽  
The Way

The object of the study of the knowledge of the variety of the Quranic reading  is the  Qur'an itself. The focus is on the difference of the reading and its articulation. The method is based on the riwayat or narration which is originated from the Prophet (Rasulullah saw) and its use is to be one of the instruments to keep the originality of the Qur’an. The validity of the reading the Qur’an is to be judged based on the valid chain  (sanad ¡a¥ī¥)  in accord with the Rasm U£mānÄ« as well as with the  Arabic grammar. Whereas the qualification of its originality is divided into six stages as follow: the first is mutawātir, the second is masyhÅ«r, the third is āhād, the fourth is syaz, the fifth is maudū‘, and the six is mudraj. Of this six catagories, the readings which can be included in the catagory of mutawātir are Qiraat Sab‘ah (the seven readings) and Qiraat ‘Asyrah  (the ten readings). To study this knowledge of reading the Qur’an (ilmu qiraat), one is advised to know about special terms being used such as  qiraat  (readings), riwayat (narration), tarÄ«q (the way), wajh (aspect), mÄ«m jama‘, sukÅ«n mÄ«m jama‘ and many others.

scholarly journals Characters of irrelevant deformations

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Shouvik Datta ◽  
Yunfeng Jiang
Keyword(s):  
Deformation Parameter ◽  
Central Charge ◽  
Scaling Limit ◽  
Small Deformation ◽  
State Dependent ◽  
Dependent Change ◽  
Left And Right ◽  
Double Scaling Limit

Abstract We analyse the $$ T\overline{T} $$ T T ¯ deformation of 2d CFTs in a special double-scaling limit, of large central charge and small deformation parameter. In particular, we derive closed formulae for the deformation of the product of left and right moving CFT characters on the torus. It is shown that the 1/c contribution takes the same form as that of a CFT, but with rescalings of the modular parameter reflecting a state-dependent change of coordinates. We also extend the analysis for more general deformations that involve $$ T\overline{T} $$ T T ¯ , $$ J\overline{T} $$ J T ¯ and $$ T\overline{J} $$ T J ¯ simultaneously. We comment on the implications of our results for holographic proposals of irrelevant deformations.

How to Distinguish Aristotle's Virtues

Phronesis ◽  
2002 ◽  
Vol 47 (2) ◽  
pp. 101-126 ◽  
Author(s):  
Marguerite Deslauriers
Keyword(s):  
Intellectual Virtue ◽  
The Political ◽  
Political Art ◽  
Intellectual Virtues ◽  
Moral Virtues ◽  
The Third ◽  
The Way

AbstractThis paper considers the distinctions Aristotle draws (1) between the intellectual virtue of phronêsis and the moral virtues and (2) among the moral virtues, in light of his commitment to the reciprocity of the virtues. I argue that Aristotle takes the intellectual virtues to be numerically distinct hexeis from the moral virtues. By contrast, I argue, he treats the moral virtues as numerically one hexis, although he allows that they are many hexeis 'in being'. The paper has three parts. In the first, I set out Aristotle's account of the structure of the faculties of the soul, and determine that desire is a distinct faculty. The rationality of a desire is not then a question of whether or not the faculty that produces that desire is rational, but rather a question of whether or not the object of the desire is good. In the second section I show that the reciprocity of phronêsis and the moral virtues requires this structure of the faculties. In the third section I show that the way in which Aristotle distinguishes the faculties requires that we individuate moral virtues according to the objects of the desires that enter into a given virtue, and with reference to the circumstances in which these desires are generated. I then explore what it might mean for the moral virtues to be different in being but not in number, given the way in which the moral virtues are individuated. I argue that Aristotle takes phronêsis and the political art to be a numerical unity in a particular way, and that he suggests that the moral virtues are, by analogy, the same kind of unity.

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