Even-visiting random walks: Exact and asymptotic results in one dimension

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 one to obtain explicit asymptotic estimates for the number of closed walks of 4k steps up to an absolute prefactor of the order of 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.