Even-visiting random walks: exact and asymptotic results in one dimension - Archive ouverte HAL
Article Dans Une Revue Journal of Physics A: Mathematical and Theoretical Année : 2001

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

M. Bauer
Denis Bernard
J. M. Luck

Résumé

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.

Dates et versions

hal-00021797 , version 1 (26-03-2006)

Identifiants

Citer

M. Bauer, Denis Bernard, J. M. Luck. Even-visiting random walks: exact and asymptotic results in one dimension. Journal of Physics A: Mathematical and Theoretical, 2001, 34, pp.2659-2679. ⟨10.1088/0305-4470/34/13/301⟩. ⟨hal-00021797⟩
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