Asymptotic analysis of the Hermite polynomials from their differential–difference equation

We propose a random matrix theory for QCD in three dimensions with a Chern-Simons term at level k which spontaneously breaks the flavor symmetry according to U. This random matrix model is obtained by adding a complex part to the action for the k = 0 random matrix model. We derive the pattern of spontaneous symmetry breaking from the analytical solution of the model. Additionally, we obtain explicit analytical results for the spectral density and the spectral correlation functions for the Dirac operator at finite matrix dimension, that become complex. In the microscopic domain where the matrix size tends to infinity, they are expected to be universal, and give an exact analytical prediction to the spectral properties of the Dirac operator in the presence of a Chern-Simons term. Here, we calculate the microscopic spectral density. It shows exponentially large (complex) oscillations which cancel the phase of the k = 0 theory. arXiv:1904.03274v1 [hep-th] 5 Apr 2019 8 The relation between the number of flavorsÑ f in [47]) and our choice N f isÑ f = 2N f . 9 Matrix models having a similar structure were studied in [68][69][70][71].

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“…This is the leading order term in Eq. (89). The oscillatory part which becomes dominant for |k| > k c can be obtained from Eq.…”

Section: Unquenched Microscopic Level Densitymentioning

confidence: 99%

Random matrix approach to three-dimensional QCD with a Chern-Simons term

Kanazawa

Kieburg

Verbaarschot

2019

J. High Energ. Phys.

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“…To calculate the explicit form of the contributions, our main tool is the Plancherel-Rotach asymptotic formula [20,4] …”

Section: Density Of Real Eigenvaluesmentioning

confidence: 99%

Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble

Forrester¹,

Nagao²

2008

J. Phys. A: Math. Theor.

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The partly symmetric real Ginibre ensemble consists of matrices formed as linear combinations of real symmetric and real anti-symmetric Gaussian random matrices. Such matrices typically have both real and complex eigenvalues. For a fixed number of real eigenvalues, an earlier work has given the explicit form of the joint eigenvalue probability density function. We use this to derive a Pfaffian formula for the corresponding summed up generalized partition function. This Pfaffian formula allows the probability that there are exactly k eigenvalues to be written as a determinant with explicit entries. It can be used too to give the explicit form of the correlation functions, provided certain skew orthogonal polynomials are computed. This task is accomplished in terms of Hermite polynomials, and allows us to proceed to analyze various scaling limits of the correlations, including that in which the matrices are only weakly nonsymmetric.

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“…For |x| < √ 2n (i.e. T > T c ), the Hermite polynomials H n (x) have the asymptotic representation (Theorem 5 in [33])…”

Section: E Asymptotic Behavior Of B Kmentioning

confidence: 99%

Critical point signatures in the cluster expansion in fugacities

et al. 2020

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The QCD baryon number density can formally be expanded into a Laurent series in fugacity, which is a relativistic generalization of Mayer's cluster expansion. We determine properties of the cluster expansion in a model with a phase transition and a critical point at finite baryon density, in which the Fourier coefficients of the expansion can be determined explicitly and to arbitrary order. The asymptotic behavior of Fourier coefficients changes qualitatively as one traverses the critical temperature and it is connected to the branching points of a thermodynamic potential associated with the phase transition. The results are discussed in the context of lattice QCD simulations at imaginary chemical potential. We argue that the location of a branching point closest to the imaginary chemical potential axis can be extracted through an analysis of an exponential suppression of Fourier coefficients. This is illustrated using the four leading coefficients both in a toy model as well as by using recent lattice QCD data.

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Asymptotic analysis of the Hermite polynomials from their differential–difference equation

Abstract: We analyze the Hermite polynomials H n (x) and their zeros asymptotically, as n → ∞. We obtain asymptotic approximations from the differential-difference equation which they satisfy, using the ray method. We give numerical examples showing the accuracy of our formulas.

Cited by 33 publications

References 34 publications

Random matrix approach to three-dimensional QCD with a Chern-Simons term

Random matrix approach to three-dimensional QCD with a Chern-Simons term

Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble

Critical point signatures in the cluster expansion in fugacities

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