Average persistence in random walks

Details Average persistence in random walks Average persistence in random walks

1998-08-26

arxiv.org/abs/cond-mat/9808290v1

Europhys. Lett. 45, 673 (1999)

Physics

Condensed Matter

Disordered Systems and Neural Networks

6 pages LaTeX, 3 postscript figures included

Scientific paper

10.1209/epl/i1999-00220-2

We study the first passage time properties of an integrated Brownian curve both in homogeneous and disordered environments. In a disordered medium we relate the scaling properties of this center of mass persistence of a random walker to the average persistence, the latter being the probability P_pr(t) that the expectation value of the walker's position after time t has not returned to the initial value. The average persistence is then connected to the statistics of extreme events of homogeneous random walks which can be computed exactly for moderate system sizes. As a result we obtain a logarithmic dependence P_pr(t)~{ln(t)}^theta' with a new exponent theta'=0.191+/-0.002. We note on a complete correspondence between the average persistence of random walks and the magnetization autocorrelation function of the transverse-field Ising chain, in the homogeneous and disordered case.

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