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

TL;DR

This paper investigates Griffiths singularities in disordered many-particle systems, revealing a universal extreme value distribution for inverse time-scales across different models and dimensions, linked to strong disorder fixed points.

Contribution

It demonstrates the universality of the inverse time-scale distribution in disordered systems with Griffiths singularities, connecting it to extreme value statistics and strong disorder renormalization group theory.

Findings

Distribution P(t^{-1},L) depends on u=t^{-1}L^{z/d}

Distribution follows the Frechet extreme value distribution

Universal behavior across different models and dimensions

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, t^{-1}, in finite samples of linear size, L, either exactly or numerically. In all cases, having a discrete symmetry, the distribution function, P(t^{-1},L), is found to depend on the variable, u=t^{-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 Frechet distribution of P(t^{-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.