  To describe the non-equilibrium dynamics of random systems, we have recently
introduced (C. Monthus and T. Garel, arxiv:0802.2502) a 'strong disorder
renormalization' (RG) procedure in configuration space that can be defined for
any master equation. In the present paper, we analyze in details the properties
of the large time dynamics whenever the RG flow is towards some "infinite
disorder" fixed point, where the width of the renormalized barriers
distribution grows indefinitely upon iteration. In particular, we show how the
strong disorder RG rules can be then simplified while keeping their asymptotic
exactness, because the preferred exit channel out of a given renormalized
valley typically dominates asymptotically over the other exit channels. We
explain why the present approach is an explicit construction in favor of the
droplet scaling picture where the dynamics is governed by the logarithmic
growth of the coherence length $l(t) \sim (\ln t)^{1/\psi}$, and where the
statistics of barriers corresponds to a very strong hierarchy of valleys within
valleys. As an example of application, we have followed numerically the RG flow
for the case of a directed polymer in a two-dimensional random medium. The full
RG rules are used to check that the RG flow is towards some infinite disorder
fixed point, whereas the simplified RG rules allow to study bigger sizes and to
estimate the barrier exponent $\psi$ of the fixed point.
