  We reconsider the problem of even-visiting random walks in one dimension.
This problem is mapped onto a non-Hermitian Anderson model with binary
disorder. We develop very efficient numerical tools to enumerate and
characterize even-visiting walks. The number of closed walks is obtained as an
exact integer up to 1828 steps, i.e., some $10^{535}$ walks. On the analytical
side, the concepts and techniques of one-dimensional disordered systems allow
to obtain explicit asymptotic estimates for the number of closed walks of $4k$
steps up to an absolute prefactor of order unity, which is determined
numerically. All the cumulants of the maximum height reached by such walks are
shown to grow as $k^{1/3}$, with exactly known prefactors. These results
illustrate the tight relationship between even-visiting walks, trapping models,
and the Lifshitz tails of disordered electron or phonon spectra.
