  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 <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 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.
