Eulerian Walkers as a model of Self-Organised Criticality

TL;DR

This paper introduces a novel self-organized criticality model where particles move on a lattice guided by arrow directions, with dynamics that generate critical states and relate to the Abelian Sandpile model.

Contribution

It presents a new model of self-organized criticality with exact analysis of its critical states and exponents, linking it to the Abelian Sandpile model.

Findings

Operators form an abelian group similar to the Sandpile model

Critical steady state characterized exactly

Critical exponents determined analytically

Abstract

We propose a new model of self-organized criticality. A particle is dropped at random on a lattice and moves along directions specified by arrows at each site. As it moves, it changes the direction of the arrows according to fixed rules. On closed graphs these walks generate Euler circuits. On open graphs, the particle eventually leaves the system, and a new particle is then added. The operators corresponding to particle addition generate an abelian group, same as the group for the Abelian Sandpile model on the graph. We determine the critical steady state and some critical exponents exactly, using this equivalence.