Analytical results for random walk persistence

Details Analytical results for random walk persistence Analytical results for random walk persistence

1998-10-12

arxiv.org/abs/cond-mat/9810136v2

Phys. Rev. E 61, 1258 (2000)

Physics

Condensed Matter

Statistical Mechanics

23 pages; accepted for publication to Phys. Rev. E (Dec. 98)

Scientific paper

10.1103/PhysRevE.61.1258

In this paper, we present the detailed calculation of the persistence exponent $\theta$ for a nearly-Markovian Gaussian process $X(t)$, a problem initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. New resummed perturbative and non-perturbative expressions for $\theta$ are obtained, which suggest a connection with the result of the alternative independent interval approximation (IIA). The perturbation theory is extended to the calculation of $\theta$ for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and non-perturbative expressions for the persistence exponent $\theta(X_0)$, describing the probability that the process remains bigger than $X_0\sqrt{}$.

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