The planar approximation. II

Itzykson, C.; Zuber, Jean-Bernard

doi:10.1063/1.524438

Cited by 907 publications

(894 citation statements)

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“…We note that the set of equations (2.10), (2.11) and (2.21) are now invariant under r → 1 r and ǫ → −ǫ, which permutes the two energy levels in H and thus interchanges g N and g M . With these observation it is straightforward to see that (2.13) becomes 22) with the same a and b as before. Thus, (2.15) and (2.16) finally become…”

Section: A Diagrammatic Approachmentioning

confidence: 66%

Renormalizing rectangles and other topics in random matrix theory

Feinberg

Zee

1997

J Stat Phys

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We consider random Hermitian matrices made of complex or real M × N rectangular blocks, where the blocks are drawn from various ensembles. These matrices have N pairs of opposite real nonvanishing eigenvalues, as well as M − N zero eigenvalues (for M > N .) These zero eigenvalues are "kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large N, M limit, in which the "rectangularity" r = M N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large N renormalization techniques. In addition to the kinematical δ-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r − 1| is held fixed as N → ∞, the N non-zero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the non-zero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As r → 1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that r → 1 drives a cross over to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

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Section: A Diagrammatic Approachmentioning

confidence: 66%

Renormalizing rectangles and other topics in random matrix theory

Feinberg

Zee

1997

J Stat Phys

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“…This equation can be found in [7], but it can already be extracted from [2]; see also [8] for a combinatorial proof.…”

Section: Expansion Around 0 Of the Hciz Integralmentioning

confidence: 99%

“…There are many ways to evaluate this integral [1,2,4,11,12,13]; the result depends only on the eigenvalues a i of A and b i of B, and is given by…”

Section: Introduction and Definitionsmentioning

confidence: 99%

“…The Harish Chandra-Itzykson-Zuber integral [1,2] plays a key role in matrix models [2,3,4,5,6], in particular in relation to (discretized) two-dimensional quantum gravity. It also displays interesting connections [7,8] with free probability theory [9,10] where A and B are N × N matrices and dΩ is the normalized Haar measure on U (N ).…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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HCIZ integral and 2D Toda lattice hierarchy

Zinn-Justin¹

2002

Nuclear Physics B

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The expression of the large $N$ Harish Chandra--Itzykson--Zuber (HCIZ) integral in terms of the moments of the two matrices is investigated using an auxiliary unitary two-matrix model, the associated biorthogonal polynomials and integrable hierarchy. We find that the large $N$ HCIZ integral is governed by the dispersionless Toda lattice hierarchy and derive its string equation. We use this to obtain various exact results on its expansion in powers of the moments.Comment: 20 pages. various minor improvements/correction

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“…In these variables, the integrand in (1) is Gaussian in X − , which can hence be integrated out, leaving us with an integral over just two matrices. Upon gauge-fixing the U(N)-symmetry, we can then carry out the integral over the unitary group using the well-known result [14]. Denoting the respective eigenvalues of X + and X 3 by x i + and x i 3 , i = 1 .…”

Section: Discrete Solutionmentioning

confidence: 99%

Boundary States of the Potts Model on Random Planar Maps

Niedner

Atkin

Wheater

2015

Springer Proceedings in Physics

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We revisit the 3-states Potts model on random planar triangulations as a Hermitian matrix model. As a novelty, we obtain an algebraic curve which encodes the partition function on the disc with both fixed and mixed spin boundary conditions. We investigate the critical behaviour of this model and find scaling exponents consistent with previous literature. We argue that the conformal field theory that describes the double scaling limit is Liouville quantum gravity coupled to the (A 4 , D 4 ) minimal model with extended W 3 -symmetry.

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The planar approximation. II

Abstract: The planar approximation is reconsidered. It is shown that a saddle point method is ineffective, due to the large number of degrees of freedom. The problem of eliminating angular variables is illustrated on a simple model coupling two N×N matrices.

Cited by 907 publications

References 6 publications

Renormalizing rectangles and other topics in random matrix theory

Renormalizing rectangles and other topics in random matrix theory

HCIZ integral and 2D Toda lattice hierarchy

Boundary States of the Potts Model on Random Planar Maps

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