Strong Griffiths singularities in random systemsand their relation to extreme value statistics
We consider interacting many particle systems with quenched disorder having strong Griffiths singularities, which are characterized by the dynamical exponent, $z$, such as random quantum systems and exclusion processes. In several $d=1$ and $d=2$ dimensional problems we have calculated the inverse time-scales, $tau^{-1}$, in finite samples of linear size, $L$, either exactly or numerically. In all cases, having a discrete symmetry, the distribution function, $P(tau^{-1},L)$, is found to depend on the variable, $u=tau^{-1}L^{z/d}$, and to be universal given by the limit distribution of extremes of independent and identically distributed random numbers. This finding is explained in the framework of a strong disorder renormalization group approach when, after fast degrees of freedom are decimated out the system is transformed into a set of non-interacting localized excitations.
The Fréchet distribution of $P(tau^{-1},L)$ is expected to hold for all random systems having a strong disorder fixed point, in which the Griffiths singularities are dominated by disorder fluctuations.
Research Institute for Solid State Physics and Optics, Budapest
