  Using exact expressions for the persistence probability and for the leading
eigenvalue of the Focker-Planck operator of a random walk in a random
environment we establish a fundamental relation between the statistical
properties of anomalous diffusion and the critical and off-critical behavior of
random quantum spin chains. Many new exact results are obtained from this
correspondence including the space and time correlations of surviving random
walks and the distribution of the gaps of the corresponding Focker-Planck
operator. In turn we derive analytically the dynamical exponent of the random
transverse-field Ising spin chain in the Griffiths-McCoy region.
