The replica limit of unitary 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].

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Unitary integrals and related matrix models

10.1093/oxfordhb/9780198744191.013.17 ◽

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.

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

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2879 ◽

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.

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UNITARY MATRIX INTEGRALS IN THE FRAMEWORK OF THE GENERALIZED KONTSEVICH MODEL

International Journal of Modern Physics A ◽

10.1142/s0217751x96002339 ◽

1996 ◽

Vol 11 (28) ◽

pp. 5031-5080 ◽

Cited By ~ 53

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.

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Jucys–Murphy Elements and Unitary Matrix Integrals

International Mathematics Research Notices ◽

10.1093/imrn/rnr267 ◽

2012 ◽

Vol 2013 (2) ◽

pp. 362-397 ◽

Cited By ~ 15

Author(s):

Sho Matsumoto ◽

Jonathan Novak

Keyword(s):

Unitary Matrix ◽

Matrix Integrals

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DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

International Journal of Quantum Information ◽

10.1142/s0219749913500159 ◽

2013 ◽

Vol 11 (01) ◽

pp. 1350015 ◽

Cited By ~ 8

Author(s):

CHI-KWONG LI ◽

REBECCA ROBERTS ◽

XIAOYAN YIN

Keyword(s):

General Scheme ◽

Quantum Gate ◽

Quantum Gates ◽

Unitary Matrix ◽

Additional Structure ◽

Unitary Matrices ◽

Decomposition Scheme ◽

Single Qubit ◽

Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

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Uniqueness Analysis of Non-Unitary Matrix Joint Diagonalization

IEEE Transactions on Signal Processing ◽

10.1109/tsp.2013.2242065 ◽

2013 ◽

Vol 61 (7) ◽

pp. 1786-1796 ◽

Cited By ~ 9

Author(s):

Martin Kleinsteuber ◽

Hao Shen

Keyword(s):

Unitary Matrix ◽

Joint Diagonalization

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Multiple phases in a generalized Gross-Witten-Wadia matrix model

Journal of High Energy Physics ◽

10.1007/jhep09(2020)081 ◽

2020 ◽

Vol 2020 (9) ◽

Cited By ~ 2

Author(s):

Jorge G. Russo ◽

Miguel Tierz

Keyword(s):

Partition Function ◽

Matrix Models ◽

Characteristic Polynomial ◽

Matrix Model ◽

Phase Structure ◽

Unitary Matrix ◽

Two Dimensional ◽

Toeplitz Determinants ◽

Dimensional Parameter ◽

Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

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A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽

10.3390/sym13071134 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1134

Author(s):

Kenta Higuchi ◽

Takashi Komatsu ◽

Norio Konno ◽

Hisashi Morioka ◽

Etsuo Segawa

Keyword(s):

Unit Circle ◽

Stationary State ◽

Discrete Time ◽

Quantum Walk ◽

Time Limit ◽

The Other ◽

Unitary Matrix ◽

Initial State ◽

Time Quantum ◽

Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

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The large-N limit of the 4d $$ \mathcal{N} $$ = 1 superconformal index

Journal of High Energy Physics ◽

10.1007/jhep11(2020)150 ◽

2020 ◽

Vol 2020 (11) ◽

Author(s):

Alejandro Cabo-Bizet ◽

Davide Cassani ◽

Dario Martelli ◽

Sameer Murthy

Keyword(s):

Effective Action ◽

Black Hole Solution ◽

Saddle Points ◽

Unitary Matrix ◽

Finite Abelian Groups ◽

Superconformal Index ◽

Cubic Form ◽

Chemical Potentials ◽

Large N ◽

Superconformal Theories

Abstract We systematically analyze the large-N limit of the superconformal index of $$ \mathcal{N} $$ N = 1 superconformal theories having a quiver description. The index of these theories is known in terms of unitary matrix integrals, which we calculate using the recently-developed technique of elliptic extension. This technique allows us to easily evaluate the integral as a sum over saddle points of an effective action in the limit where the rank of the gauge group is infinite. For a generic quiver theory under consideration, we find a special family of saddles whose effective action takes a universal form controlled by the anomaly coefficients of the theory. This family includes the known supersymmetric black hole solution in the holographically dual AdS5 theories. We then analyze the index refined by turning on flavor chemical potentials. We show that, for a certain range of chemical potentials, the effective action again takes a universal cubic form that is controlled by the anomaly coefficients of the theory. Finally, we present a large class of solutions to the saddle-point equations which are labelled by group homomorphisms of finite abelian groups of order N into the torus.

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