Renormalization group solution of the Chutes & Ladder model

• Model of biased persistent random walk with exact renormalization group solution.
• Non-universal exponent and de-localization transition.
• Pedagogical example of transport in a complex network.
• Strategy for foraging behavior or for “lifting” of Markov chains.

We analyze a semi-infinite one-dimensional random walk process with a biased motion that is incremental in one direction and long-range in the other. On a network with a fixed hierarchy of long-range jumps, we find with exact renormalization group calculations that there is a dynamical transition between a localized adsorption phase and an anomalous diffusion phase in which the mean-square displacement exponent depends non-universally on the Bernoulli coin. We relate these results to similar findings of unconventional phase behavior in hierarchical networks.