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

M. Bauer·D. Bernard·J. M. Luck
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Abstract

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.

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M. Bauer, D. Bernard, J. M. Luck. 2001-02-28. Even-visiting random walks: exact and asymptotic results in one dimension. https://doi.org/10.1088/0305-4470%2F34%2F13%2F301

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