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

Akemann, Gernot; Cikovic, Milan; Venker, Martin

doi:10.1007/s00220-018-3201-1

Cited by 35 publications

(52 citation statements)

References 34 publications

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“…For local universality in standard FTE at the soft edge see [45]. While all the quoted results apply to real eigenvalues in Hermitian FTE, very recently universality was proven for the complex eigenvalues of a generalised FTE in the bulk of the spectrum, both at strong and weak non-Hermiticity [15]. While it is known that in the bulk the rate of convergence is exponentially fast for the Ginibre ensemble, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…For a single ensemble of complex non-Hermitian matrices without constraint, universality has been studied rigorously by several authors [16,17,54,37]. Adding a constraint, the ensembles become nondeterminantal, and universality still holds [15]. What can be said about the universality of products of non-Gaussian random matrices?…”

Section: Introductionmentioning

confidence: 99%

Products of random matrices from fixed trace and induced Ginibre ensembles

Akemann¹,

Cikovic²

2018

J. Phys. A: Math. Theor.

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“…To get an impression of the τ -dependence of the kernel, we consider the origin z = 0. Here, we can use the relations ωω = 1 and ω/ω = −1 from (5.4), to obtain 17) with the relation between v and τ from (5.4). It is not justified to take the weak non-Hermiticity limit (N → ∞ with the scaling (4.1)) of (5.15) because of the restriction (5.5) being violated, which was crucial for our analysis above.…”

Section: (512)mentioning

confidence: 99%

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

Nagao

Akemann

Kieburg

et al. 2020

J. Phys. A: Math. Theor.

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We investigate a one-parameter family of Coulomb gases in two dimensions, which are confined to an ellipse due to a hard wall constraint, and are subject to an additional external potential. At inverse temperature β = 2 we can use the technique of planar orthogonal polynomials, borrowed from random matrix theory, to explicitly determine all k-point correlation functions for a fixed number of particles N . These are given by the determinant of the kernel of the corresponding orthogonal polynomials, which in our case are the Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials, depending on the choice of external potential, as shown in a companion paper recently published by three of the authors. In the rotationally invariant case, when the ellipse becomes the unit disc, our findings agree with that of the ensemble of truncated unitary random matrices. The thermodynamical large-N limit is investigated in the local scaling regime in the bulk and at the edge of the spectrum at weak and strong non-Hermiticity. We find new universality classes in these limits and recover the sine-and Bessel-kernel in the Hermitian limit. The limiting global correlation functions of particles in the interior of the ellipse are more difficult to obtain but found in the special cases corresponding to the Chebyshev polynomials.

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“…Therefore, taking expectation values, using the definition according to (2), and letting t → ∞, we obtain…”

Section: Dynamics For 1d Coulomb Gasesmentioning

confidence: 99%

“…Proof Suppose that the point ζ ∈ C. Recall that R N,k is the k-point correlation function (2) for the 2D Coulomb gas given by (1). Let ψ ∈ C ∞ 0 (C) be an arbitrary test function.…”

Section: Ward Identities In 2dmentioning

confidence: 99%

The High Temperature Crossover for General 2D Coulomb Gases

Akemann

Byun

2019

J Stat Phys

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We consider N particles in the plane influenced by a general external potential that are subject to the Coulomb interaction in two dimensions at inverse temperature β . At large temperature, when scaling β = 2c/N with some fixed constant c > 0, in the large-N limit we observe a crossover from Ginibre's circular law or its generalization to the density of non-interacting particles at β = 0. Using several different methods we derive a partial differential equation of generalized Liouville type for the crossover density. For radially symmetric potentials we present some asymptotic results and give examples for the numerical solution of the crossover density. These findings generalise previous results when the interacting particles are confined to the real line. In that situation we derive an integral equation for the resolvent valid for a general potential and present the analytic solution for the density in case of a Gaussian plus logarithmic potential.

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

Cited by 35 publications

References 34 publications

Products of random matrices from fixed trace and induced Ginibre ensembles

Products of random matrices from fixed trace and induced Ginibre ensembles

Families of two-dimensional Coulomb gases on an ellipse: correlation functions and universality

The High Temperature Crossover for General 2D Coulomb Gases

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