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

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

2001-02-28

arxiv.org/abs/cond-mat/0102512v1

Journal of Physics A 34 (2001), 2659.

Physics

Condensed Matter

Statistical Mechanics

24 pages, 4 figures. To appear in J. Phys. A

Scientific paper

10.1088/0305-4470/34/13/301

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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