Average persistence of random walks

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 pr(t) that the expectation value x(t) 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 pr(t) ~ ln (t)− with a new exponent = 0.191 ± 0.002. 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.