Scaling and Fractal formation in Persistence

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.