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We study random XY and (dimerized) XX spin-1/2 quantum spin chains at theirquantum phase transition driven by the anisotropy and dimerization,respectively. Using exact expressions for magnetization, correlation functionsand energy gap, obtained by the free fermion technique, the critical andoff-critical (Griffiths-McCoy) singularities are related to persistenceproperties of random walks. In this way we determine exactly the decayexponents for surface and bulk transverse and longitudinal correlations,correlation length exponent and dynamical exponent.Comment: 4 pages RevTeX, 1 eps-figure include

Absence of dynamical crossover in the vortex creep near by the second peak effect in superconducting Hg-1201 single crystals