The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

Akemann, Gernot; Phillips, M. J.

doi:10.1007/s10955-014-0962-6

Cited by 19 publications

(24 citation statements)

References 49 publications

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“…For this model, the local correlation kernel at the right/left endpoint of the spectrum was derived in the regime of weak non-Hermiticity when 1 − τ = O(N −1/3 ). The kernel (2.7) was derived there as well from the large argument limit of such an intermediate process, see [6,Eq.(4.28)].…”

Section: Resultsmentioning

confidence: 99%

Scaling Limits of Planar Symplectic Ensembles

Akemann,

Byun,

Kang

2021

Preprint

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We consider various asymptotic scaling limits N → ∞ for the 2N complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with a Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials Q(ζ) = |ζ| 2λ − (2c/N ) log |ζ|, with λ > 0 and c > −1, the limiting kernel obeys a linear differential equation of fractional order 1/λ at the origin. For integer m = 1/λ it can be solved in terms of Mittag-Leffler functions. In the last part, we derive Ward's equation for a general class of potentials as a tool to investigate universality. This allows us to determine the functional form of kernels that are translation invariant up to its integration domain.

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Section: Resultsmentioning

confidence: 99%

Scaling Limits of Planar Symplectic Ensembles

Akemann,

Byun,

Kang

2021

Preprint

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“…This oneparameter family of random matrices interpolates between the Gaussian Orthogonal Ensemble of real symmetric matrices (GOE, τ = 1) and real Ginibre ensemble of fully asymmetric matrices (rGinE, τ = 0), see [35] for discussions. Both rGinE and its one-parameter extension (9) have enjoyed considerable interest in the literature in recent years [36,37,38,39,40].…”

Section: Resultsmentioning

confidence: 99%

Nonlinear analogue of the May−Wigner instability transition

Fyodorov

Khoruzhenko

2016

Proc. Natl. Acad. Sci. U.S.A.

121

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We study a system of N ≫ 1 degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate μ. We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically nontrivial regime characterized by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May−Wigner instability transition originally discovered by local linear stability analysis.complex systems | equilibrium | model ecosystems | random matrices W ill diversity make a food chain more or less stable? The prevailing view in the midtwentieth century was that diverse ecosystems have greater resilience to recover from events displacing the system from equilibrium and hence are more stable. This "ecological intuition" was challenged by Robert May in 1972 (1). At that time, computer simulations suggested that large complex systems assembled at random might become unstable as the system complexity increases (2). May's 1972 paper complemented that work with an analytic investigation of the neighborhood stability of a model ecosystem whereby N species at equilibrium are subject to random interactions.The time evolution of large complex systems, of which model ecosystems is one example, is often described within the general mathematical framework of coupled first-order nonlinear ordinary differential equations (ODEs). In the context of generic systems, the Hartman−Grobner theorem then asserts that the neighborhood stability of a typical equilibrium can be studied by replacing the nonlinear interaction functions near the equilibrium with their linear approximations. It is along these lines that May suggested looking at the linear modelto study the stability of large complex systems. Here J = ðJ jk Þ is the coupling matrix and μ > 0. In the absence of interactions, i.e., when all J jk = 0, system [1] is self-regulating: If disturbed from the equilibrium y 1 = y 2 = . . . = y N = 0, it returns back with some characteristic relaxation time set by μ. In an ecological context, y j ðtÞ is interpreted as the variation about the equilibrium value, y j = 0, in the population density of species j at time t. The element J jk of the coupling matrix J, which is known as the community matrix in ecology, measures the per capita effect of species k on species j at the presumed equilibrium. Generically, the community matrix is asymmetric, J jk ≠ J kj . For complex multispecies systems, information about the interaction between species is rarely available at the level of detail sufficient for the exact computation of the community matrix and a subsequent stability analysis. Instead, May considered an ensemble of community matrices J assembled at random, whereby ...

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“…More recently, various other non-Hermitian ensembles have attracted interest (see [1,24,35,34,3] for a small selection). One particular categorization relevant to the present work is the 'geometrical triumvirate' of ensembles described in [36,31,38,17], which identifies random matrix ensembles with the three classical surfaces of constant curvature: the plane, the sphere and the pseudo-or anti-sphere.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…where χ φ is the indicator function, and S A is the spherical annulus corresponding to the boundary circles in the complex plane with radii (22). We also suspect that the single-ring theorem of [15] can be generalized to the case here, that is, the polynomial V in (5) can perhaps be broadened to include the logarithmic expressions we find in (3). Another outstanding calculation is that of the average over the product of characteristic polynomials (18).…”

Section: Further Workmentioning

confidence: 99%

An induced real quaternion spherical ensemble of random matrices

Mays

Ponsaing

2017

Random Matrices: Theory Appl.

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We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a 2 × 2 complex matrix). We define the ensemble by the matrix probability distribution function that is proportional toThese matrices can also be constructed via a procedure called 'inducing', using a product of a Wishart matrix (with parameters n, N ) and a rectangular Ginibre matrix of size (N +L)×N . The inducing procedure imposes a repulsion of eigenvalues from 0 and ∞ in the complex plane, with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of L, n and N . By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue m-point correlation functions, and in particular the eigenvalue density (given by m = 1).We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of n and L. After a stereographic projection the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behaviour of the density near the real line based on analogous results in the β = 1 and β = 2 ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the β = 4 induced spherical ensemble.

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The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

Cited by 19 publications

References 49 publications

Scaling Limits of Planar Symplectic Ensembles

Scaling Limits of Planar Symplectic Ensembles

Nonlinear analogue of the May−Wigner instability transition

An induced real quaternion spherical ensemble of random matrices

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