Persistence in higher dimensions : a finite size scaling study

Abstract

We show that the persistence probability $P(t,L)$, in a coarsening system of linear size $L$ at a time $t$, has the finite size scaling form $P(t,L)\sim L^{-z\hat{I}¸}f(\frac{t}{L^z})$ where $\hat{I}¸$ is the persistence exponent and $z$ is the coarsening exponent. The scaling function $f(x)\sim x^{-\hat{I}¸}$ for $x\ll 1$ and is constant for large $x$. The scaling form implies a fractal distribution of persistent sites with power-law spatial correlations. We study the scaling numerically for Glauber-Ising model at dimension $d=1$ to 4 and extend the study to the diffusion problem. Our finite size scaling ansatz is satisfied in all these cases providing a good estimate of the exponent $\hat{I}¸$.

4 pages in RevTeX with 6 figures. To appear in Phys. Rev. E