Non-Hermitian tridiagonal random matrices and returns to the origin of a random walk

We study a class of tridiagonal matrix models, the "q-roots of unity" models. which includes the sign (q = 2) and the clock (q = infinity) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of M-k are integers that count closed random walks on the line such that each site is visited a number of times multiple of q. We obtain an explicit evaluation for them.