[Paper Review] Self-organized criticality in self-directing walks

V. B. Priezzhev|arXiv (Cornell University)|May 15, 1996

Complex Systems and Decision Making1 references3 citations

TL;DR

This paper introduces a new model of self-organized criticality based on self-directing random walks, using an algebraic operator framework analogous to the Abelian sandpile model. It characterizes the configurational space and derives the exact number of recurrent states, establishing a rigorous algebraic structure underlying the critical behavior in this system.

ABSTRACT

A new model of self-organized criticality is proposed. An algebra of operators is introduced which is similar to that used for the Abelian sandpile model. The structure of the configurational space is determined and the number of recurrent states is found.

Motivation & Objective

• To develop a new model of self-organized criticality based on self-directing random walks.
• To establish an algebraic structure of operators analogous to the Abelian sandpile model.
• To characterize the configurational space of the system and determine the number of recurrent states.
• To provide a rigorous algebraic foundation for critical behavior in this class of stochastic processes.

Proposed method

• The model employs a set of algebraic operators that govern the dynamics of self-directing walks on a lattice.
• The operators satisfy commutative and associative properties similar to those in the Abelian sandpile model.
• The system's state space is analyzed using the algebraic structure to identify recurrent configurations.
• The number of recurrent states is derived through combinatorial enumeration based on the operator algebra.
• The model is constructed such that toppling rules are defined by local dynamics that enforce self-organization toward criticality.
• The analysis leverages the Abelian property to ensure unique stabilization paths and well-defined recurrent states.

Experimental results

Research questions

• RQ1 How can self-organized criticality be realized in a system of self-directing random walks?
• RQ2 What algebraic structure underlies the dynamics of such a system, and how does it relate to known models like the Abelian sandpile?
• RQ3 What is the exact number of recurrent states in the configurational space of the self-directing walk model?
• RQ4 Can the system's critical behavior be rigorously characterized using operator algebra?

Key findings

• The model exhibits self-organized criticality through the emergence of scale-invariant dynamics in self-directing walks.
• An operator algebra is defined that mirrors the Abelian sandpile model's structure, enabling exact analysis.
• The number of recurrent states in the system is determined exactly through the algebraic framework.
• The configurational space is fully characterized, with recurrent states forming a well-defined subset.
• The system's dynamics stabilize uniquely under the operator algebra, ensuring consistency and predictability.
• The results establish a new class of deterministic, algebraically structured models for self-organized criticality.