Non-equilibrium dynamics of disordered systems: renormalization flow towards an ‘infinitedisorder’ fixed point at large times

Abstract

To describe the non-equilibrium dynamics of random systems, we have recently introduced(Monthus and Garel, 2008 J. Phys. A: Math. Theor. 41 255002) a ‘strong disorderrenormalization’ (RG) procedure in configuration space that can be defined for any masterequation. In the present paper, we analyze in detail the properties of the large timedynamics whenever the RG flow is towards some ‘infinite disorder’ fixed point, where thewidth of the renormalized barrier distribution grows indefinitely upon iteration. Inparticular, we show how the strong disorder RG rules can be then simplified while keepingtheir asymptotic exactness, because the preferred exit channel out of a given renormalizedvalley typically dominates asymptotically over the other exit channels. We explain why thepresent approach is an explicit construction favoring the droplet scaling picturewhere the dynamics is governed by the logarithmic growth of the coherence lengthl(t)∼(lnt)1/ψ, and where the statistics of barriers corresponds to a very strong hierarchy of valleyswithin valleys. As an example of application, we have followed numerically the RG flow forthe case of a directed polymer in a two-dimensional random medium. The full RG rules areused to check that the RG flow is towards some infinite disorder fixed point, whereas thesimplified RG rules allow us to study bigger sizes and to estimate the barrier exponentψ of the fixed point.