Level spacings for weakly asymmetric real random matrices and application to two-color QCD with chemical potential

Bloch, Jacques; Bruckmann, Falk; Meyer, Nils; Schierenberg, Sebastian

doi:10.1007/jhep08(2012)066

Cited by 6 publications

(7 citation statements)

References 61 publications

(82 reference statements)

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“…In investigating the (de)localization of the eigenstates, Feinberg and Zee [10], argued that imaginary eigenvalues near the real axis can attract when perturbed by a Hermitian matrix by providing a 2 × 2 example of an imaginary diagonal matrix perturbed by a 2 × 2 Hermitian matrix with zero diagonal entries. Later, Bloch et al [15] considered antisymmetric perturbations of real symmetric matrices in the context of two-color quantum chromodynamics and provided examples that a Hermitian matrix perturbed by a real antisymmetric perturbation can give rise to attraction of eigenvalues. To our knowledge, attraction of the eigenvalues and their eventual aggregation on the real line, in a general setting, has not been proved.…”

Section: A Backgroundmentioning

confidence: 99%

“…Later, Bloch et al [15] considered antisymmetric perturbations of real symmetric matrices in the context of two-color quantum chromodynamics and provided examples that a Hermitian matrix perturbed by a real antisymmetric perturbation can give rise to attraction of eigenvalues. To our knowledge, attraction of the eigenvalues and their eventual aggregation on the real line, in a general setting, has not been proved.…”

Section: Introductionmentioning

confidence: 99%

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Eigenvalue Attraction

Movassagh¹

2015

J Stat Phys

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We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.

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Section: A Backgroundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Eigenvalue Attraction

Movassagh¹

2015

J Stat Phys

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“…As our method adopts the level spacing and the smallest eigenvalue distributions that are extremely sensitive to the fitting parameter as primary fitting observables (see Refs. [36][37][38] for recent efforts along this line), it enjoys a clear advantage over the methods using n-level correlation functions, and presents promising applications in analyzing the numerical data of QCD-like theories.…”

Section: Introductionmentioning

confidence: 99%

Universality crossover between chiral random matrix ensembles and twisted SU(2) lattice Dirac spectra

Nishigaki¹

2012

Phys. Rev. D

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Motivated by the statistical fluctuation of Dirac spectrum of QCD-like theories subjected to (pseudo)reality-violating perturbations and in the ε regime, we compute the smallest eigenvalue distribution and the level spacing distribution of chiral and nonchiral parametric random matrix ensembles of Dyson-Mehta-Pandey type. To this end we employ the Nyström-type method to numerically evaluate the Fredholm Pfaffian of the integral kernel for the chG(O,S)E-chGUE and G(O,S)E-GUE crossover. We confirm the validity and universality of our results by comparing them with several lattice models, namely fundamental and adjoint staggered Dirac spectra of SU(2) quenched lattice gauge theory under the twisted boundary condition (imaginary chemical potential) or perturbed by phase noise. Both in the zero-virtuality region and in the spectral bulk, excellent one-parameter fitting is achieved already on a small 4 4 lattice. Anticipated scaling of the fitting parameter with the twisting phase, mean level spacing, and the system size allows for precise determination of the pion decay (diffusion) constant F in the low-energy effective Lagrangian.2 boundary condition or imaginary chemical potential) [19][20][21]. These cases are distinct from the previously mentioned case in that the Dirac eigenvalues stay real even after the inclusion of symmetry violations and their statistical behavior exhibits crossover, rather than develops into the complex plane. In terms of the effective σ-model description, these two cases are almost identical, save for the difference of the sign of trBQ †B Q term (µ 2 F 2 or (iµ) 2 F 2 , see Sec. IV.).Crossover between universality classes of Hermitian random matrix ensembles [22][23][24], namely GOE-GUE and GSE-GUE, is extensively studied in the context of disordered [25,26] and quantum-chaotic Hamiltonians [27][28][29] with its time-reversal invariance slightly broken by weak magnetic field or AB flux applied [30]. On the other hand, the chiral or superconducting variant of universality crossover appears to be a relatively unexplored field so far. Previous attempts in this area either focused on the level number variance and the spectral form factor (both of which are integral transforms of the two-level correlator and insensitive to chiralness) of two-color QCD with AB fluxes versus GOE-GUE crossover [19], or have fruited in a series of tours de force by Damgaard and collaborators [31,32] devoted to the analytical computation of the level density and individual small eigenvalue distributions for the spectral crossover within the chiral Gaussian unitary ensemble (chGUE) class, due to the imaginary isospin chemical potential. Despite that analytic results for the microscopic spectral correlation functions are known for some time for chGOE-chGUE and chGSE-chGUE crossover [33,34], 1 they are yet to encounter with physical application. To the best of our knowledge, the only example of the crossover involving different Hermitian chiral universality classes discussed in a physical setting is the CI-C transition ...

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“…The effect of this is best understood by first considering the impact on eigenvalues of the opposite transformation: gradually adding off-diagonal elements to a diagonal matrix. A simple perturbation analysis (e.g., Bloch et al 2012), strictly valid only when the off-diagonal entries of G are small compared to the diagonal entries, already gives the general trends, which can be confirmed numerically for larger off-diagonal entries: When exploitative interaction, where G ij and G ji (i = j) have opposite signs, dominate (as for direct predator-prey interactions), the off-diagonal entries lead to a narrowing of the eigenvalue distribution along the real axis. The general expectation, e.g., with statistically unrelated or positively correlated G ij and G ji , however, is that the distribution will be broadened, see, e.g.…”

Section: S7 Effects Of Structural Instability Of the Management Modelmentioning

confidence: 99%

Moving towards ecosystem-based fisheries management: Options for parameterizing multi-species biological reference points

Moffitt

Punt

Holsman

et al. 2016

Deep Sea Research Part II: Topical Studies in Oceanography

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The case of fisheries management illustrates how the inherent structural instability of ecosystems can have deep-running policy implications. We contrast ten types of management plans to achieve maximum sustainable yields (MSY) from multiple stocks and compare their effectiveness based on a management strategy evaluation (MSE) that uses complex food webs in its operating model. Plans that target specific stock sizes (B MSY ) consistently led to higher yields than plans targeting specific fishing pressures (F MSY ). A new self-optimising control rule, introduced here for its robustness to structural instability, led to intermediate yields. Most plans outperformed single-species management plans with pressure targets set without considering multispecies interactions. However, more refined plans to "maximise the yield from each stock separately", in the sense of a Nash equilibrium, produced total yields comparable to plans aiming to maximise total harvested biomass, and were more robust to structural instability. Our analyses highlight trade-offs between yields, amenability to negotiations, pressures on biodiversity, and continuity with current approaches in the European context. Based on these results, we recommend directions for developments of EU fisheries policy.

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Level spacings for weakly asymmetric real random matrices and application to two-color QCD with chemical potential

Cited by 6 publications

References 61 publications

Eigenvalue Attraction

Eigenvalue Attraction

Universality crossover between chiral random matrix ensembles and twisted SU(2) lattice Dirac spectra

Moving towards ecosystem-based fisheries management: Options for parameterizing multi-species biological reference points

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