Dynamical Crossover in Invasion Percolation

The dynamical properties of the invasion percolation on the square lattice are investigated with emphasis on the geometrical properties on the growing cluster of infected sites. The exterior frontier of this cluster forms a critical loop ensemble (CLE), whose length \((l)\), the radius \((r)\) and also roughness \((w)\) fulfill the finite size scaling hypothesis. The dynamical fractal dimension of the CLE defined as the exponent of the scaling relation between \(l\) and \(r\) is estimated to be \(D_f=1.81\pm0.02\). By studying the autocorrelation functions of these quantities we show importantly that there is a crossover between two time regimes, in which these functions change behavior from power-law at the small times, to exponential decay at long times. In the vicinity of this crossover time, these functions are estimated by log-normal functions. We also show that the increments of the considered statistical quantities, which are related to the random forces governing the dynamics of the observables undergo an anticorrelation/correlation transition at the time that the crossover takes place.