Characteristic polynomials of random banded Hessenberg matrices and Hermite-Pad\'e approximation

We consider a class of random banded Hessenberg matrices with independent entries having identical distributions along diagonals. The distributions may be different for entries belonging to different diagonals. For a sequence of $n\times n$ matrices in the class considered, we investigate the asymptotic behavior of their empirical spectral distribution as $n$ tends to infinity.Comment: Minor changes in this version, including some added references, 20 page