A Stochastic Description for Extremal Dynamics

TL;DR

This paper models extremal dynamics using the Linear Fractional Stable Motion (LFSM), providing analytical and numerical evidence, and derives exact correlation functions and fractional equations for interface growth.

Contribution

It introduces LFSM as a comprehensive stochastic model for extremal dynamics, linking it to spatio-temporal correlations and deriving exact analytical expressions.

Findings

LFSM accurately models extremal dynamics.

Derived exact n-point correlation functions.

Established fractional order equations for interface growth.

Abstract

We show that extremal dynamics is very well modelled by the "Linear Fractional Stable Motion" (LFSM), a stochastic process entirely defined by two exponents that take into account spatio-temporal correlations in the distribution of active sites. We demonstrate this numerically and analytically using well-known properties of the LFSM. Further, we use this correspondence to write an exact expressions for an n-point correlation function as well as an equation of fractional order for interface growth in extremal dynamics.