We consider the real eigenvalues of an $(N \times N)$ elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter $\tau_N\in [0,1]$. In almost-Hermitian regime where $1-\tau_N=\Theta(N^{-1})$, we obtain large-$N$ expansion mean and variance number eigenvalues. Furthermore, derive limiting empirical distributions eigenvalues, which interpolate Wigner semicircle law u...

is the jacobian matrix of F . If this matix has n distinct eigenvalues λk (real or complex), the system can be diagonalized, and the solutions can be expressed as combinations of exponentials of the form ek. Therefore, x − x∗ → 0 if all the eigenvalues have a negative real part. If one or more eigenvalues have positive real part, |x− x∗| → ∞. Therefore, • The fixed point x∗ is stable if all the...

In the recent publication [E. Kanzieper and G. Akemann, Phys. Rev. Lett. 95, 230201 (2005)], an exact solution was reported for the probability pn,k to find exactly k real eigenvalues in the spectrum of an n × n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof...

has only real eigenvalues, which were shown to exist by Davies in [3]. In the same paper, he showed that the spectrum of −iH is equal to the set of its eigenvalues, so this implies that the spectrum is real. This was conjectured in [1], and Chugunova and Pelinovsky proved in [2] that all but finitely many eigenvalues are real, and gave numerical evidence that all eigenvalues are real. Our appro...