Optimal fluctuations and tail states of non-Hermitian operators

Marchetti, F. M.; Simons, Benjamin D.

doi:10.1088/0305-4470/34/49/305

Cited by 3 publications

(4 citation statements)

References 51 publications

(108 reference statements)

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“…Therefore, in the so-called mean-field approximation [12,36] the extension in the imaginary axis will be ξ-independent. It should be noted however that that mean-field approach would have given a constant DOS within a zone around η = 0, rather than (18).…”

Section: A Circular Complex Gaussian Potential U(t)mentioning

confidence: 99%

“…close to η = 0, the calculated DOS is no longer valid. To capture the behavior of the DOS in this region, a zero-dimensional analysis similar to [21,36,38,39] is needed.…”

Section: A Circular Complex Gaussian Potential U(t)mentioning

confidence: 99%

“…The independence of ρ from ξ is not surprising: the density of states of the Hermitian (diagonal) part of ( 2) is independent of ξ. Therefore, in the so-called mean-field approximation [12,36] the extension in the imaginary axis will be ξ-independent. It should be noted however that that mean-field approach would have given a constant DOS within a zone around η = 0, rather than (18).…”

Section: Lyapunov Exponent and Dosmentioning

confidence: 99%

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Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Kazakopoulos

Moustakas

2008

Phys. Rev. E

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We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive non-linear Schrödinger equation with a Gaussian random pulse as initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We also calculate the distribution of a set of scattering data of the Zakharov-Shabat operator that determine the asymptotics of the eigenfunctions. We analyze two cases, one with circularly symmetric complex Gaussian pulses and the other with real Gaussian pulses. We discuss the implications in the context of information transmission through non-linear optical fibers.

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Section: A Circular Complex Gaussian Potential U(t)mentioning

confidence: 99%

“…close to η = 0, the calculated DOS is no longer valid. To capture the behavior of the DOS in this region, a zero-dimensional analysis similar to [21,36,38,39] is needed.…”

Section: A Circular Complex Gaussian Potential U(t)mentioning

confidence: 99%

Section: Lyapunov Exponent and Dosmentioning

confidence: 99%

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Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Kazakopoulos

Moustakas

2008

Phys. Rev. E

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“…Halperin's method [31], which works so nice for the hermitian/dark soliton case, fails here because of non-hermiticity. Non-hermitian random operators have received considerable attention in the literature (see [34] for a list of important references on the subject). They have a wide range of applications, in non-equilibrium statistical mechanics, random classical dynamics, the physics of polymers, QCD, neural networks, and, in the case at hand, soliton physics and communications.…”

Section: Bright Solitonsmentioning

confidence: 99%

Transmission of information via the non-linear Scroedinger equation: The random Gaussian input case

Kazakopoulos¹,

Moustakas²

2012

Preprint

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The explosion of demand for ultra-high information transmission rates over the last decade has necessitated the usage of increasingly high light intensities for fiber optical transmissions. As a result, the fiber non-linearities need to be treated non-perturbatively. Similar analyses in the past have focused on the effects of non-linearities on existing transmission technologies, e.g. WDM.In this paper we take advantage of the fact that, under certain assumptions, light transmission through optical fibers can be described using the non-linear Schroedinger equation, which is exactly integrable. As a particular example, we show that in the low Gaussian noise limit, the Gaussian input distribution has a higher mutual information than the transmission using WDM over the same available bandwidth.

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Random matrices close to Hermitian or unitary: overview of methods and results

Fyodorov¹,

Sommers²

2003

J. Phys. A: Math. Gen.

165

237

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Abstract. The paper discusses recent progress in understanding statistical properties of eigenvalues of (weakly) non-Hermitian and non-unitary random matrices. The first type of ensembles is of the formĴ =Ĥ − iΓ, withĤ being a large random N × N Hermitian matrix with independent entries "deformed" by a certain anti-Hermitian N × N matrix iΓ satisfying in the limit of large dimension N the condition: TrĤ 2 ∝ N TrΓ 2 . HereΓ can be either a random or just a fixed given Hermitian matrix. Ensembles of such a type withΓ ≥ 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics. Related models are used to mimic complex spectra of the Dirac operator with chemical potential in the context of Quantum Chromodynamics.Ensembles of the second type, arising naturally in scattering theory of discrete-time systems, are formed by N × N matricesÂ with complex entries such thatÂ †Â =Î −T . ForT = 0 this coincides with the Circular Unitary Ensemble, and 0 ≤T ≤Î describes deviation from unitarity. Our result amounts to answering statistically the following old question: given the singular values of a matrixÂ describe the locus of its eigenvalues.We systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non-trivial crossover from WignerDyson statistics of real/unimodular eigenvalues typical for Hermitian/unitary matrices to Ginibre statistics in the complex plane typical for ensembles with strong non-Hermiticity: < TrĤ 2 >∝< TrΓ 2 > when N → ∞. Finally we discuss (scarce) results available on eigenvector statistics for weakly nonHermitian random matrices. IntroductionAs is well-known, the statistics of highly excited bound states of closed quantum chaotic systems of quite different microscopic nature is universal. Namely, it turns out to be independent of the microscopic details when sampled on energy intervals large in comparison with the mean level spacing ∆, but smaller than the so called Thouless energy scale. The latter is related by the Heisenberg uncertainty principle to the relaxation time necessary for the classically chaotic system to reach equilibrium in phase space [1]. Moreover, the spectral correlation functions turn out to be exactly those which are provided by the theory of large random Hermitian matrices with independent, identically distributed Gaussian entries. The correspondence holds in the limit of large matrix dimension on the so-called local scale. The Random matrices close to Hermitian or unitary2 local scale is determined by the typical separation ∆ = X i − X i−1 between neighbouring eigenvalues situated around a point X, with the brackets standing for the statistical averaging. Microscopic arguments supporting the use of random matrices for describing the universal properties of quantum chaotic systems have been provided recently by several groups, based both on traditional semiclassical periodic orbit expansions [2,3] and on advanced field-theoretical methods [4,5].In paralle...

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Optimal fluctuations and tail states of non-Hermitian operators

Cited by 3 publications

References 51 publications

Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Nonlinear Schrödinger equation with random Gaussian input: Distribution of inverse scattering data and eigenvalues

Transmission of information via the non-linear Scroedinger equation: The random Gaussian input case

Random matrices close to Hermitian or unitary: overview of methods and results

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