Strong Griffiths singularities in random systems and their relation to extreme value statistics

Abstract

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, ${\ensuremath{\tau}}^{\ensuremath{-}1}$, in finite samples of linear size, $L$, either exactly or numerically. In all cases, having a discrete symmetry, the distribution function, $P({\ensuremath{\tau}}^{\ensuremath{-}1},L)$, is found to depend on the variable, $u={\ensuremath{\tau}}^{\ensuremath{-}1}{L}^{z}$, 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 noninteracting localized excitations. The Fr\'echet distribution of $P({\ensuremath{\tau}}^{\ensuremath{-}1},L)$ is expected to hold for all random systems having a strong disorder fixed point, in which the Griffiths singularities are dominated by disorder fluctuations.

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• Physical Review B - View