Smallest eigenvalue distribution of the fixed-trace Laguerre beta-ensemble

Chen, Yang; Liu, Dang-Zheng; Zhou, Da-Sheng

doi:10.1088/1751-8113/43/31/315303

Cited by 31 publications

(57 citation statements)

References 20 publications

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“…[7] for earlier heuristic arguments. Local statistics at the soft and hard edge have been analyzed for fixed trace β-ensembles in [37,14,30]. More general fixed trace models have been considered where the trace of a polynomial in the random matrix is fixed [5].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble

Akemann

Cikovic

Venker

2018

Commun. Math. Phys.

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We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace Tr(XX * ) of the matrix X with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications and interpolates between the celebrated sine and Ginibre kernel. Our results constitute a first proof of universality of the interpolating kernel. Furthermore, in the limit of strong non-Hermiticity, where the support of the limiting measure remains an ellipse, we obtain local Ginibre statistics in the bulk of the spectrum.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble

Akemann

Cikovic

Venker

2018

Commun. Math. Phys.

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“…They allow to quantify the degree of entanglement of |ϕ by looking at the smallest (or largest) eigenvalue only. It distribution is known to be universal, agreeing with that of the Wishart-Laguerre ensemble without constraint [22]. In contrast, when the Hamiltonians are non-Hermitian, we keep the bases of left and right eigenvectors instead, as typically operators acting on |ϕ will yield complex eigenvalues.…”

Section: Introductionmentioning

confidence: 97%

Products of random matrices from fixed trace and induced Ginibre ensembles

Akemann¹,

Cikovic²

2018

J. Phys. A: Math. Theor.

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We investigate the microcanonical version of the complex induced Ginibre ensemble, by introducing a fixed trace constraint for its second moment. Like for the canonical Ginibre ensemble, its complex eigenvalues can be interpreted as a two-dimensional Coulomb gas, which are now subject to a constraint and a modified, collective confining potential. Despite the lack of determinantal structure in this fixed trace ensemble, we compute all its density correlation functions at finite matrix size and compare to a fixed trace ensemble of normal matrices, representing a different Coulomb gas. Our main tool of investigation is the Laplace transform, that maps back the fixed trace to the induced Ginibre ensemble. Products of random matrices have been used to study the Lyapunov and stability exponents for chaotic dynamical systems, where the latter are based on the complex eigenvalues of the product matrix. Because little is known about the universality of the eigenvalue distribution of such product matrices, we then study the product of m induced Ginibre matrices with a fixed trace constraint -which are clearly non-Gaussian -and M − m such Ginibre matrices without constraint. Using an m-fold inverse Laplace transform, we obtain a concise result for the spectral density of such a mixed product matrix at finite matrix size, for arbitrary fixed m and M . Very recently local and global universality was proven by the authors and their coworker for a more general, single elliptic fixed trace ensemble in the bulk of the spectrum. Here, we argue that the spectral density of mixed products is in the same universality class as the product of M independent induced Ginibre ensembles. arXiv:1711.05488v3 [math-ph]

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“…In addition, for the regular Wishart-Laguerre ensemble we also indicate a relation between the recurrence relation and the determinantal results of Forrester and Hughes [28], and explicitly demonstrate the equivalence between the two results for rectangularity α = 0, 1. Similarly, for the fixed-trace scenario we prove the equivalence of the recursion-based expression and the result of Chen, Liu and Zhou [35] based on the inverse Laplace transform of a determinant, again for α = 0, 1.…”

Section: Introductionmentioning

confidence: 54%

“…The universality aspects of the regular Wishart-Laguerre ensemble have been explored in several notable works [29][30][31][32][33][34][38][39][40][41][55][56][57][58][59][60]. For the fixed trace case, the local statistical properties of the eigenvalues have been studied in [35,61]. In particular, it has been shown that the fixed trace and the regular Wishart-Laguerre ensembles share identical universal behaviour for large n at the hard edge, in the bulk and at the soft edge for α fixed [61].…”

Section: Large N α Evaluations and Comparison With Tracy-widom Densitymentioning

confidence: 99%

“…In figure 7 we set N 1 = 11, N 2 = 21, = 1, and examine the effect of different choices of k 1 , k 2 on one level density and smallest eigenvalue density for the coupled kicked tops. For (a), (i) we have k 1 = 0.5, k 2 = 1 for which the classical phase spaces of the individual (11,11), (11,15), (11,25), (21,21), (21,25), (21,35). tops consist mostly of regular orbits [54].…”

Section: Coupled Kicked Topsmentioning

confidence: 99%

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Smallest eigenvalue density for regular or fixed-trace complex Wishart–Laguerre ensemble and entanglement in coupled kicked tops

Kumar¹,

Sambasivam²,

Anand³

2017

J. Phys. A: Math. Theor.

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Abstract. The statistical behaviour of the smallest eigenvalue has important implications for systems which can be modeled using a Wishart-Laguerre ensemble, the regular one or the fixed trace one. For example, the density of the smallest eigenvalue of the Wishart-Laguerre ensemble plays a crucial role in characterizing multiple channel telecommunication systems. Similarly, in the quantum entanglement problem, the smallest eigenvalue of the fixed trace ensemble carries information regarding the nature of entanglement.For real Wishart-Laguerre matrices, there exists an elegant recurrence scheme suggested by Edelman to directly obtain the exact expression for the smallest eigenvalue density. In the case of complex Wishart-Laguerre matrices, for finding exact and explicit expressions for the smallest eigenvalue density, existing results based on determinants become impractical when the determinants involve large-size matrices. In this work, we derive a recurrence scheme for the complex case which is analogous to that of Edelman's for the real case. This is used to obtain exact results for the smallest eigenvalue density for both the regular, and the fixed trace complex Wishart-Laguerre ensembles. We validate our analytical results using Monte Carlo simulations. We also study scaled Wishart-Laguerre ensemble and investigate its efficacy in approximating the fixed-trace ensemble. Eventually, we apply our result for the fixed-trace ensemble to investigate the behaviour of the smallest eigenvalue in the paradigmatic system of coupled kicked tops.

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Smallest eigenvalue distribution of the fixed-trace Laguerre beta-ensemble

Cited by 31 publications

References 20 publications

Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble

Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble

Products of random matrices from fixed trace and induced Ginibre ensembles

Smallest eigenvalue density for regular or fixed-trace complex Wishart–Laguerre ensemble and entanglement in coupled kicked tops

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