Eigenvalue curves of asymmetric tridiagonal random matrices. (English) Zbl 0983.82006

Goldsheid, Ilya Ya.; Khoruzhenko, Boris A.

Electron. J. Probab. 5, Paper No. 16, 28 p. (2000).

The paper gives a detailed analysis of the limiting behaviour of the spectrum of tridiagonal matrices with i.i.d. real entries and periodic boundary conditions, as the size goes to infinity. It is shown that the limit distribution is supported by curves in \(\mathbb{C}\), and equations for these curves are determined. In particular, conditions on the parameters of the distribution of the random variables are obtained to decide whether the asymptotic spectrum is real, or whether it is non-real. These problems arise from the study of discrete random Schrödinger operators with imaginary vector potentials in dimension one.

Reviewer: Hans Crauel (Ilmenau)

MSC:

82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
15B52	Random matrices (algebraic aspects)
47B80	Random linear operators
60H25	Random operators and equations (aspects of stochastic analysis)
47B39	Linear difference operators
37H15	Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents

Keywords:

random matrix; Schrödinger operator; Lyapunov exponent; eigenvalue distribution; complex eigenvalue; limiting behaviour; spectrum of tridiagonal matrices; limit distribution; asymptotic spectrum; discrete random Schrödinger operators