scholarly journals First Order Asymptotics of Matrix Integrals; A Rigorous Approach Towards the Understanding of Matrix Models

2004 ◽  
Vol 244 (3) ◽  
pp. 527-569 ◽  
Author(s):  
Alice Guionnet
Keyword(s):  
Matrix Models ◽  
First Order ◽  
Rigorous Approach ◽  
Matrix Integrals

scholarly journals UNITARY MATRIX MODELS AND 2D QUANTUM GRAVITY

1992 ◽  
Vol 07 (29) ◽  
pp. 2753-2762 ◽  
Author(s):  
S. DALLEY ◽  
C. V. JOHNSON ◽  
T. R. MORRIS ◽  
A. WÄTTERSTAM
Keyword(s):  
Matrix Models ◽  
Integrable Hierarchies ◽  
Renormalization Group Equation ◽  
Closed String ◽  
Singular Solutions ◽  
Unitary Matrix ◽  
Group Equation ◽  
Gravitational System ◽  
Large N ◽  
Matrix Integrals

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system — open-closed string theory. Non-perturbative solutions of the multicritical unitary matrix models map to non-singular solutions of the 'renormalization group' equation for the string susceptibility, [Formula: see text]. We also demonstrate that the large-N solutions of unitary matrix integrals in external fields, studied by Gross and Newman, equal the non-singular pure closed-string solutions of [Formula: see text].

Unitary integrals and related matrix models

2018 ◽  
Author(s):  
Nicolas Orantin
Keyword(s):  
Matrix Models ◽  
Unitary Matrix ◽  
Technical Tool ◽  
Basic Properties ◽  
Matrix Integrals ◽  
Unitary Model ◽  
M Theory ◽  
Pair Correlator

This article examines the basic properties of unitary matrix integrals using three matrix models: the ordinary unitary model, the Brézin-Gross-Witten (BGW) model and the Harish-Chandra-Itzykson-Zuber (HCIZ) model. The tricky sides of the story are given special attention, such as the de Wit-’t Hooft anomaly in unitary integrals and the problem of correlators with Itzykson-Zuber measure. The method of character expansions is also emphasized as a technical tool. The article first provides an overview of the theory of the BGW model, taking into account the de Wit-’t Hooft anomaly and the M-theory of matrix models, before discussing the theory of the HCIZ integral. In particular, it describes the basics of character calculus, character expansion of the HCIZ integral, character expansion for the BGW model and Leutwyler-Smilga integral, and pair correlator in HCIZ theory.

scholarly journals Unitary Matrix Integrals, Primitive Factorizations, and Jucys-Murphy Elements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sho Matsumoto ◽  
Jonathan Novak
Keyword(s):  
Symmetric Group ◽  
Matrix Models ◽  
Unitary Matrix ◽  
Group Characters ◽  
Matrix Integrals ◽  
International Audience

International audience A factorization of a permutation into transpositions is called "primitive'' if its factors are weakly ordered.We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described. Une factorisation en transpositions d'une permutation est dite "primitive'' si ses facteurs sont ordonnés. Nous discutons du problème de l'énumération des factorisations primitives de permutations, et de sa place dans la hiérarchie des problèmes de factorisation précédemment étudiés. Nous présentons plusieurs formules énumérant certaines classes de factorisations primitives,et nous soulignons des connexions intéressantes avec les éléments Jucys-Murphy, les caractères des groupes symétriques, et les modèles de matrices.

scholarly journals UNITARY MATRIX INTEGRALS IN THE FRAMEWORK OF THE GENERALIZED KONTSEVICH MODEL

1996 ◽  
Vol 11 (28) ◽  
pp. 5031-5080 ◽  
Author(s):  
A. MIRONOV ◽  
A. MOROZOV ◽  
G. W. SEMENOFF
Keyword(s):  
Partition Function ◽  
External Field ◽  
Matrix Models ◽  
Strong Coupling ◽  
Weak Coupling ◽  
Unitary Matrix ◽  
Simple Method ◽  
Generating Functional ◽  
New Approach ◽  
Matrix Integrals

We advocate a new approach to the study of unitary matrix models in external fields which emphasizes their relationship to generalized Kontsevich models (GKM's) with nonpolynomial potentials. For example, we show that the partition function of the Brezin–Gross–Witten model (BGWM), which is defined as an integral over unitary N × N matrices, [Formula: see text], can also be considered as a GKM with potential [Formula: see text]. Moreover, it can be interpreted as the generating functional for correlators in the Penner model. The strong and weak coupling phases of the BGWM are identified with the "character" (weak coupling) and "Kontsevich" (strong coupling) phases of the GKM, respectively. This type of GKM deserves classification as a p = −2 model (i.e. c = 28 or c = −2) when in the Kontsevich phase. This approach allows us to further identify the Harish-Chandra–Itzykson–Zuber integral with a peculiar GKM, which arises in the study of c = 1, theory, and, further, with a conventional two-matrix model which is rewritten in Miwa coordinates. Some further extensions of the GKM treatment which are inspired by the unitary matrix models which we have considered are also developed. In particular, as a by-product, a new, simple method of fixing the Ward identities for matrix models in an external field is presented.

scholarly journals Matrix Models and Matrix Integrals

2006 ◽  
Vol 146 (1) ◽  
pp. 63-72 ◽  
Author(s):  
A. D. Mironov
Keyword(s):  
Matrix Models ◽  
Matrix Integrals

scholarly journals ON THE COUNTING OF COLORED TANGLES

2000 ◽  
Vol 09 (08) ◽  
pp. 1127-1141 ◽  
Author(s):  
PAUL ZINN-JUSTIN ◽  
JEAN-BERNARD ZUBER
Keyword(s):  
Matrix Models ◽  
Generating Function ◽  
Closed Loop ◽  
Elliptic Functions ◽  
Vertex Model ◽  
Random Lattice ◽  
Asymptotic Behaviors ◽  
Matrix Integral ◽  
Matrix Integrals

The connection between matrix integrals and links is used to define matrix models which count alternating tangles n which each closed loop is weighted with a factor n, i.e. may be regarded as decorated with n possible colors. For n=2, the corresponding matrix integral is that recently solved in the study of the random lattice six-vertex model. The generating function of alternating 2-color tangle is provided in terms of elliptic functions, expanded to 16-th order (16 crossings) and its asymptotic behaviors is given.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Stochasting modeling of planetary ring systems

1984 ◽  
Vol 75 ◽  
pp. 461-469 ◽  
Author(s):  
Robert W. Hart
Keyword(s):  
Maximum Entropy ◽  
Ring System ◽  
The Sun ◽  
Ultimate State ◽  
Interparticle Interactions ◽  
Ring Systems ◽  
First Order ◽  
Planetary Ring ◽  
Insight Into

ABSTRACTThis paper models maximum entropy configurations of idealized gravitational ring systems. Such configurations are of interest because systems generally evolve toward an ultimate state of maximum randomness. For simplicity, attention is confined to ultimate states for which interparticle interactions are no longer of first order importance. The planets, in their orbits about the sun, are one example of such a ring system. The extent to which the present approximation yields insight into ring systems such as Saturn's is explored briefly.

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