Edge Scaling Limit of the Spectral Radius for Random Normal Matrix Ensembles at Hard Edge
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Edge Scaling Limit of the Spectral Radius for Random Normal Matrix Ensembles at Hard Edge Seong-Mi Seo1 Received: 25 May 2020 / Accepted: 9 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We investigate local statistics of eigenvalues for random normal matrices, represented as 2D determinantal Coulomb gases, in the case when the eigenvalues are forced to be in the support of the equilibrium measure associated with an external field. For radially symmetric external fields with sufficient growth at infinity, we show that the fluctuations of the spectral radius around a hard edge tend to follow an exponential distribution as the number of eigenvalues tends to infinity. As a corollary, we obtain the order statistics of the moduli of eigenvalues. Keywords Random normal matrices · Hard edge · Spectral radius · Universality · 2D Coulomb gases Mathematics Subject Classification 60B20 · 60G55 · 82B21
1 Introduction and Results In random matrix theory, there have been numerous studies of the spectral radius of large size matrices. The limiting distributions of the largest eigenvalue of classical random matrix ensembles, Gaussian orthogonal, unitary, and symplectic ensembles, were studied by Forrester [20] and Tracy-Widom [39,40]. More generally, a type of universality for Wigner random matrices was proved by Soshnikov [38]. The study of the scaling limit of correlation functions at the edge of the spectrum allowed to prove the universality of the largest eigenvalue distribution for some invariant ensembles [17]. The edge behavior of the spectrum of a random normal matrix is different from that of a random hermitian matrix, which is expressed in terms of Painlevé II. In the random normal matrix model, one considers random normal matrices of size n with a probability measure of the form
Communicated by Abhishek Dhar.
B 1
Seong-Mi Seo [email protected] School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu Seoul 02455, Republic of Korea
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S.-M. Seo
dPn (M) =
1 Zn
e−ntr Q(M) dM,
(1.1)
where Q : C → R ∪ {+∞} is an external potential, d M is the surface measure on complexvalued n ×n matrices M with M ∗ M = M M ∗ induced from the standard metric on Cn×n , and Zn is a normalizing constant. The system of eigenvalues is represented as a 2D Coulomb gas model (one-component plasma) at a specific temperature, subjected to the external potential Q. If the external potential grows sufficiently fast at infinity, the eigenvalues accumulate on a compact set called the droplet as the size of matrix n goes to infinity. In the case of Q(z) = |z|2 , the system of eigenvalues is represented by the complex Ginibre ensemble [23] and the droplet is the closed unit disk. The maximum of the moduli of the eigenvalues for the complex Ginibre ensemble has the same distribution as that of a set of independent random variables [28], and with a proper scaling, the limit law of its spectral edge follows the Gumbel distribution [34]. This result has been generalized to a
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