Renormalization Group Solution of the Chutes&Ladder Model

TL;DR

This paper uses renormalization group methods to analyze a hierarchical random walk model, revealing a phase transition between localized and anomalous diffusion phases with non-universal behavior.

Contribution

It introduces an exact renormalization group approach to study a hierarchical random walk, uncovering a dynamical transition and non-universal diffusion exponents.

Findings

Identification of a phase transition between localized and anomalous diffusion phases.

Derivation of non-universal mean-square displacement exponents.

Connection of results to unconventional phase behavior in hierarchical networks.

Abstract

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.