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.

中文翻译：

---

入侵渗透中的动态交叉

研究了方形格子上入侵渗透的动力学特性，重点是不断增长的感染位点簇的几何特性。该集群的外部边界形成了一个临界环系综 (CLE)，其长度 $(l)$、半径 $(r)$ 以及粗糙度 $(w)$ 满足有限尺寸缩放假设。CLE 的动态分形维数定义为 $l$ 和 $r$ 之间的比例关系的指数，估计为 $D_f=1.81\pm0.02$。通过研究这些量的自相关函数，我们重要地表明两个时间体系之间存在交叉，其中这些函数从小时间的幂律变化到长时间的指数衰减。在此交叉时间附近，这些函数由对数正态函数估计。