Gaussian ensemble of tridiagonal symmetric random matrices

Abstract

We obtained exact energy level correlators for the Gaussian ensemble of finite tridiagonal symmetric matrices by employing the connection between the linear eigenvalue problem and periodic Toda equations. Solutions of the latter help us to parametrise matrices in the ensemble, which is equivalent to the decomposition onto the spectral and rotational degrees of freedom in the theory of filled random matrices. The rotational variables can be integrated out reducing expressions for energy level correlators to multidimensional integrals over eigenvalues only. We found that density of states for the considered ensemble does not have a semicircle shape as N → ∞. The spectral statistics approaches the Poisson type with one singular point in the same limit.