  We study the random XY spin chain in a transverse field by analyzing the
susceptibility of the ground state fidelity, numerically evaluated through a
standard mapping of the model onto quasi-free fermions. It is found that the
fidelity susceptibility and its scaling properties provide useful information
about the phase diagram. In particular it is possible to determine the Ising
critical line and the Griffiths phase regions, in agreement with previous
analytical and numerical results.
