The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system

@article{Baik2018TheLR,
  title={The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov–Shabat system},
  author={Jinho Baik and Thomas Bothner},
  journal={The Annals of Applied Probability},
  year={2018}
}
The real Ginibre ensemble consists of $n\times n$ real matrices ${\bf X}$ whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius $R_n=\max_{1\leq j\leq n}|z_j({\bf X})|$ of the eigenvalues $z_j({\bf X})\in\mathbb{C}$ of a real Ginibre matrix ${\bf X}$ follows a different limiting law (as $n\rightarrow\infty$) for $z_j({\bf X… 

Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices

It has been known since the pioneering paper of Mark Kac, that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac' approach by

On the solution of the Zakharov-Shabat system, which arises in the analysis of the largest real eigenvalue in the real Ginibre ensemble

Let λmax be a shifted maximal real eigenvalue of a random N×N matrix with independent N(0, 1) entries (the ‘real Ginibre matrix’) in the N → ∞ limit. It was shown by Poplavskyi, Tribe, Zaboronski [9]

Edge Distribution of Thinned Real Eigenvalues in the Real Ginibre Ensemble

This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed

Fluctuations and correlations for products of real asymmetric random matrices

. We study the real eigenvalue statistics of products of independent real Ginibre random matrices. These are matrices all of whose entries are real i.i.d. standard Gaussian random variables. For such

Malnormal matrices

We exhibit an operator norm bounded, infinite sequence $\{A_n\}$ of $4n \times 4n$ complex matrices for which the commutator map $X\mapsto XA_n - A_nX$ is uniformly bounded below as an operator over

Real eigenvalues of elliptic random matrices

We consider the real eigenvalues of an (N ×N) real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter τN ∈ [0, 1]. In the almost-Hermitian regime where 1 − τN =

From Painlevé to Zakharov–Shabat and beyond: Fredholm determinants and integro-differential hierarchies

As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This

Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel

We rigorously compute the integrable system for the limiting (N →∞) distribution function of the extreme momentum of N noninteracting fermions when confined to an anharmonic trap V (q) = q2n for n ∈

On the number of real eigenvalues of a product of truncated orthogonal random matrices

Let O be chosen uniformly at random from the group of ( N + L ) × ( N + L ) orthogonal matrices. Denote by ˜ O the upper-left N × N corner of O , which we refer to as a truncation of O . In this

Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems

Motivated by the phenomenon of duality for interacting particle systems we introduce two classes of Pfaffian kernels describing a number of Pfaffian point processes in the ‘bulk’ and at the ‘edge’.

References

SHOWING 1-10 OF 37 REFERENCES

On the distribution of the largest real eigenvalue for the real Ginibre ensemble

Let $\sqrt{N}+\lambda_{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). We study the large deviations behaviour of

Direct and inverse scattering on the line

Part I. The Forward Problem: Distinguished solutions Fundamental matrices Fundamental tensors Behavior of fundamental tensors as $|x|\rightarrow\infty$ the Functions $\Delta_k$ Behavior of

Large Deformations of the Tracy–Widom Distribution I: Non-oscillatory Asymptotics

We analyze the left-tail asymptotics of deformed Tracy–Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft

Fredholm determinants, differential equations and matrix models

AbstractOrthogonal polynomial random matrix models ofN×N hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (ϕ(x)ψ(y)−ψ(x)ϕ(y))/x−y. This paper is

Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble

It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled t → ∞ ?> limit of the annihilation process A + A → ∅ ?> .

Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations

Author(s): Dieng, Momar | Abstract: We derive Painlev #x27;e--type expressions for the distribution of the $m^{th}$ largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge

Eigenvalue statistics of the real Ginibre ensemble.

A computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented, relevant to May's stability analysis of biological webs.

Large n Limit of Gaussian Random Matrices with External Source, Part I

AbstractWe consider the random matrix ensemble with an external source defined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the

Log-Gases and Random Matrices

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments,

Statistical Ensembles of Complex, Quaternion, and Real Matrices

Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the