  We study random XY and (dimerized) XX spin-1/2 quantum spin chains at their
quantum phase transition driven by the anisotropy and dimerization,
respectively. Using exact expressions for magnetization, correlation functions
and energy gap, obtained by the free fermion technique, the critical and
off-critical (Griffiths-McCoy) singularities are related to persistence
properties of random walks. In this way we determine exactly the decay
exponents for surface and bulk transverse and longitudinal correlations,
correlation length exponent and dynamical exponent.
