Reinforced walks in two and three dimensions

TL;DR

This paper investigates reinforced random walks in two and three dimensions, using simulations and heuristic analysis to explore phase transitions and universality classes, finding that the transition occurs at zero reinforcement and that models share similar critical behavior.

Contribution

The study demonstrates that site and bond reinforced walks belong to the same universality class and that the phase transition occurs at zero reinforcement, challenging previous numerical suggestions.

Findings

Site and bond reinforcement models share the same universality class.

The phase transition in 2D occurs at zero reinforcement.

Quasi-static approximation accurately predicts large-time scaling.

Abstract

In probability theory, reinforced walks are random walks on a lattice (or more generally a graph) that preferentially revisit neighboring `locations' (sites or bonds) that have been visited before. In this paper, we consider walks with one-step reinforcement, where one preferentially \emph{revisits} locations irrespective of the number of visits. Previous numerical simulations [A. Ordemann {\it et al.}, Phys. Rev. E {\bf 64}, 046117 (2001)] suggested that the site model on the lattice shows a phase transition at finite reinforcement between a random-walk like and a collapsed phase, in both 2 and 3 dimensions. The very different mathematical structure of bond and site models might also suggest different phenomenology (critical properties, etc.). We use high statistics simulations and heuristic arguments to suggest that site and bond reinforcement are in the same universality class, and that the purported phase transition in 2 dimensions actually occurs at zero coupling constant. We also show that a quasi-static approximation predicts the large time scaling of the end-to-end distance in the collapsed phase of both site and bond reinforcement models, in excellent agreement with simulation results.