# Bulk universality for generalized Wigner matrices

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We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is...

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We consider N Ã— N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away...

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The random matrix theory is used to bridge the network structures and the dynamical processes defined on them. We propose a possible dynamical mechanism for the enhancement effect of network structures on synchronization processes, based upon which a dynamic-based index of the synchronizability...

- On the eigenproblems of PT-symmetric oscillators. Shin, K. C. // Journal of Mathematical Physics;Jun2001, Vol. 42 Issue 6
We consider the non-Hermitian Hamiltonian H=-d[sup 2]/dx[sup 2]+P(x[sup 2])-(ix)[sup 2n+1] on the real line, where P(x) is a polynomial of degree at most n>=1 with all non-negative real coefficients (possibly Pâ‰¡0). It is proved that the eigenvalues Î» must be in the sector |arg...

- Discrete Wigner function by symmetric informationally complete positive operator valued measure. Bar-on, T. // Journal of Mathematical Physics;Jul2009, Vol. 50 Issue 7, p072106
We construct a version of the discrete Wigner function making use of symmetric informationally complete positive operator valued measure. We will show that this version is the natural discrete analog of continuous Wigner function. In addition, to this discrete Wigner function has many properties...

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We obtain uniform asymptotics for polynomials orthogonal on a fixed and varying arc of the unit circle with a positive analytic weight function. We also complete the proof of the large s asymptotic expansion for the Fredholm determinant with the kernel sinz/(Ï€ z) on the interval [0,s],...

- The Spectrum of Heavy Tailed Random Matrices. Arous, Gérard; Guionnet, Alice // Communications in Mathematical Physics;Mar2008, Vol. 278 Issue 3, p715
Let X N be an N â†’ N random symmetric matrix with independent equidistributed entries. If the law P of the entries has a finite second moment, it was shown by Wigner [14] that the empirical distribution of the eigenvalues of X N , once renormalized by $$\sqrt{N}$$ , converges almost...

- A new family of integrable models in (2+1) dimensions associated with Hermitian symmetric spaces. Strachan, I. A. B. // Journal of Mathematical Physics;Jul92, Vol. 33 Issue 7, p2477
In a series of papers Fordy and his collaborators have studied families of integrable models in (1+1) dimensions associated with Hermitian symmetric spaces. These models were also generalized to (2+1) dimensions. However, the generalization used is not unique, and in this paper a different...