  In random quantum magnets, like the random transverse Ising chain, the low
energy excitations are localized in rare regions and there are only weak
correlations between them. It is a fascinating question whether these
correlations are completely irrelevant in the sense of the renormalization
group. To answer this question, we calculate the distribution of the excitation
energy of the random transverse Ising chain in the disordered Griffiths phase
with high numerical precision by the strong disorder renormalization group
method and - for shorter chains - by free-fermion techniques. Asymptotically,
the two methods give identical results, which are well fitted by the Fr\'echet
limit law of the extremes of independent and identically distributed random
numbers. Given the finite size corrections, the two numerical methods give very
similar results, but they differ from the correction term for uncorrelated
random variables. This fact shows that the weak correlations between low-energy
excitations in random quantum magnets are not entirely irrelevant.
