# 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}
}
• Published 7 August 2018
• Mathematics
• The Annals of Applied Probability
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 $z_j({\bf X… ## Figures and Tables from this paper • Mathematics Electronic Journal of Probability • 2020 It has been known since the pioneering paper of Mark Kac, that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac' approach by Let λmax be a shifted maximal real eigenvalue of a random N×N matrix with independent N(0, 1) entries (the ‘real Ginibre matrix’) in the N → ∞ limit. It was shown by Poplavskyi, Tribe, Zaboronski [9] • Mathematics Annales Henri Poincaré • 2022 This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed • Mathematics, Physics • 2021 . We study the real eigenvalue statistics of products of independent real Ginibre random matrices. These are matrices all of whose entries are real i.i.d. standard Gaussian random variables. For such • Mathematics • 2020 We exhibit an operator norm bounded, infinite sequence$\{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 • Mathematics • 2021 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 = 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 • Physics Annales de l'Institut Henri Poincaré, Probabilités et Statistiques • 2022 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 ∈ • Mathematics Electronic Journal of Probability • 2022 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 • Mathematics The Annals of Probability • 2022 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 • Mathematics • 2016 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 • Mathematics • 1988 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 • Mathematics • 2017 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 • Mathematics • 1994 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 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 → ∅ ?> . 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
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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.
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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
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 with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the