Scaling and Fractal formation in Persistence

TL;DR

This paper investigates the fractal nature and scaling behavior of persistent sites in a one-dimensional reaction-diffusion model, revealing relationships between fractal dimension, dynamical, and persistence exponents.

Contribution

It introduces a numerical study of unvisited sites in the A+A→∅ model, establishing a relation among key exponents and discussing initial condition effects.

Findings

Unvisited sites form fractals below a certain length scale.

The exponents are sensitive to initial particle density.

A possible crossover behavior at late times is discussed.

Abstract

The spatial distribution of unvisited/persistent sites in $d=1$ $A+A\to\emptyset$ model is studied numerically. Over length scales smaller than a cut-off $\xi(t)\sim t^{z}$, the set of unvisited sites is found to be a fractal. The fractal dimension $d_{f}$, dynamical exponent $z$ and persistence exponent $\theta$ are related through $z(1-d_{f})=\theta$. The observed values of $d_{f}$ and $z$ are found to be sensitive to the initial density of particles. We argue that this may be due to the existence of two competing length scales, and discuss the possibility of a crossover at late times.