Random Walk Approach to Simple Evolution Model

TL;DR

This paper models the avalanche width evolution in a self-organized criticality system using a random walk approach, deriving critical exponents and identifying the critical point with numerical support.

Contribution

It introduces a random walk framework to analyze avalanche dynamics and determines critical exponents and the critical point in the evolution model.

Findings

Critical exponents match previous mean field results.

SOC appears at a critical parameter value of 2/3.

Numerical studies of continuous time random walks support the model.

Abstract

The dynamics of the avalanche width in the evolution model is described using a random walk picture. In this approach the critical exponents for avalanche distribution, $\tau$, and avalanche average time, $\gamma$, are found to be the same as in the previous mean field approximation but SOC appear at $\lambda_{critical}=2/3$, which is very close to numerical value. A continuous time random walk is studied numerically as a possible way to reconstruct in simpler concepts the evolution model.