Circular Law

Girko, V. L.

doi:10.1137/1129095

Cited by 311 publications

(316 citation statements)

References 5 publications

Supporting

Mentioning

306

Contrasting

Order By: Relevance

“…The circular law for N × N matrices with i.i.d Gaussian entries says that, when normalized by N −1/2 the density of eigenvalues becomes uniform on the unit disk as N → ∞ (See [15] for a proof of this fact when the entries are i.i.d. Gaussian, and [2] and [24] for more general results).…”

Section: In the Bulkmentioning

confidence: 99%

“…It is known (see [15] [2] [9] [8]) that the density functions of real and complex eigenvalues is approximately constant on (− √ N , √ N ) and the disc of radius √ N respectively (here N is the size of the matrices). We study the local correlations of eigenvalues in four different regions: near a real point u √ N with −1 < u < 1 (real bulk), near ± √ N (real edge), near a (non-real) complex point u √ N with |u| < √ N (complex bulk), and near u √ N with |u| = √ N , Im(u) = 0 (complex edge).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

Borodin

Sinclair

2009

Commun. Math. Phys.

134

218

View full text Add to dashboard Cite

show abstract

Section: In the Bulkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

Borodin

Sinclair

2009

Commun. Math. Phys.

134

218

View full text Add to dashboard Cite

show abstract

“…It would be interesting to see if the results reported in this paper can be extended, at least qualitatively, to the 2D-OCP in more generic confining potentials [14,42,61] (other than radially symmetric [3,34,36]) or for the plasma on different planar surfaces (e.g. on a cylinder [11,12]).…”

Section: Discussionmentioning

confidence: 89%

“…This distribution on the disk is known as circular law in random matrix theory [6,8,36,37]. The electrostatic energy and the entropy of the gas are then easy to compute,…”

Section: Thermodynamics Of the 2d-ocpmentioning

confidence: 99%

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

2016

View full text Add to dashboard Cite

The two-dimensional one-component plasma is an ubiquitous model for several vortex systems. For special values of the coupling constant βq 2 (where q is the particles charge and β the inverse temperature), the model also corresponds to the eigenvalues distribution of normal matrix models. Several features of the system are discussed in the limit of large number N of particles for generic values of the coupling constant. We show that the statistics of a class of radial observables produces a rich phase diagram, and their asymptotic behaviour in terms of large deviation functions is calculated explicitly, including next-to-leading terms up to order 1/N . We demonstrate a split-off phenomenon associated to atypical fluctuations of the edge density profile. We also show explicitly that a failure of the fluid phase assumption of the plasma can break a genuine 1/N -expansion of the free energy. Our findings are corroborated by numerical comparisons with exact finite-N formulae valid for βq 2 = 2.

show abstract

“…Our intuition is born from results in the random matrix theory of Girko [17], Edelman [14], and Bai [4]. The basic idea of their work is the following: given a square matrix whose elements are real random variables drawn from a distribution with a finite sixth moment, in the limit of infinite dimensions, the normalized spectrum (or eigenvalues) of the matrix will converge to a uniform distribution on the unit disk in the complex plane.…”

mentioning

confidence: 99%

Routes to chaos in high-dimensional dynamical systems: A qualitative numerical study

Albers

Sprott

2006

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

This paper examines the most probable route to chaos in high-dimensional dynamical systems in a very general computational setting. The most probable route to chaos in high-dimensional, discrete-time maps is observed to be a sequence of Neimark-Sacker bifurcations into chaos. A means for determining and understanding the degree to which the Landau-Hopf route to turbulence is non-generic in the space of C r mappings is outlined. The results comment on previous results of Newhouse, Ruelle, Takens, Broer, Chenciner, and Iooss. In their first edition of Fluid Mechanics [26], Landau and Lifschitz proposed a route to turbulence in fluid systems. Since then, much work, in dynamical systems, experimental fluid dynamics, and many other fields has been done concerning the routes to turbulence. In this paper, we present early results from the first large statistical study of the route to chaos in a very general class of high-dimensional, C r , dynamical systems. Our results contain both some reassurances based on a wealth of previous results and some surprises. We conclude that, for high-dimensional discrete-time maps, the most probable route to chaos (in our general construction) from a fixed point is via at least one Neimark-Sacker bifurcation, followed by persistent zero Lyapunov exponents, and finally a bifurcation into chaos. We observe both the Ruelle-Takens

show abstract

Circular Law

Cited by 311 publications

References 5 publications

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

Large Deviations of Radial Statistics in the Two-Dimensional One-Component Plasma

Routes to chaos in high-dimensional dynamical systems: A qualitative numerical study

Contact Info

Product

Resources

About