Average persistence of random walks

¹ NIC c/o Forschungszentrum Jülich - D-52425 Jülich, Germany Institut für Theoretische Physik, Universität zu Köln - D-50923 Köln, Germany
² Research Institute for Solid State Physics, H-1525 Budapest - P.O.Box 49, Hungary Institute for Theoretical Physics, Szeged University - H-6720 Szeged, Hungary

Received: 28 August 1998
Accepted: 11 January 1999

Abstract

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 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 with a new exponent . We note 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.